Non-polynomial q-Askey Scheme: Integral Representations, Eigenfunction Properties, and Polynomial Limits

Abstract

We construct a non-polynomial generalization of the q-Askey scheme. Whereas the elements of the q-Askey scheme are given by q-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose integrands are built from Ruijsenaars’ hyperbolic gamma function. Alternatively, the integrands can be expressed in terms of Faddeev’s quantum dilogarithm, Woronowicz’s quantum exponential, or Kurokawa’s double sine function. We present the basic properties of all the elements of the scheme, including their integral representations, joint eigenfunction properties, and polynomial limits.

1 Introduction

An important tool in the classification of the different families of classical (basic) hypergeometric orthogonal polynomials is the (q-)Askey scheme [25]. It provides a tree (drawn upside down) of the different families of (q-)orthogonal polynomials, where each downward oriented edge corresponds to an explicit limit transition from one family to another. In particular, each family in the q-Askey scheme satisfies orthogonality relations, a 3-term recurrence relation, and a second-order q-difference equation. At the top of the q-Askey scheme lies the celebrated family of Askey–Wilson (AW) polynomials [3] which additionally admits a duality relation exchanging its geometric and spectral parameter.

The theory of hypergeometric series and integrals has four levels: rational, trigonometric, hyperbolic and elliptic, with increasing complexity. Moreover, there exist formal limits of the form elliptic $\rightarrow$ hyperbolic/trigonometric $\rightarrow$ rational. The associated Gamma functions are the ordinary one (rational), the q-Gamma function (trigonometric, requiring $|q| < 1$), the hyperbolic Gamma function, and the elliptic Gamma function. The Askey scheme and the q-Askey scheme describe the rational and trigonometric levels, respectively. The q-Askey scheme has a trivial extension to the hyperbolic level, with the families now consisting of products of polynomials $P_n {\tilde{P}}_m$, where $P_n$ (resp. ${\tilde{P}}_m$) corresponds to the associated family in the q-Askey scheme relative to the base $q=e^{2i\pi b^2}$ (resp. ${\tilde{q}} = e^{2i \pi b^{-2}}$). However, in this case the orthogonality relations break down when $b \in \mathbb R$, which corresponds to the natural hyperbolic regime. Finally, there is no elliptic Askey scheme; however, there is a fairly well understood theory of elliptic hypergeometric biorthogonal functions which generalizes the classical basic hypergeometric orthogonal polynomials. For instance, it was shown in [6] that all families in the q-Askey scheme are obtained as limits of elliptic hypergeometric biorthogonal functions.

Non-polynomial extensions of (q-)Askey schemes have been less extensively studied in the literature. In particular, the first two levels of a trigonometric non-polynomial extension of the q-Askey scheme were introduced in [28, 29]. The top-level family generalizing the AW polynomials consists of a very-well poised $_8\phi _7$ basic hypergeometric series called the AW-function (see also [9, 18, 23, 28, 59]). The $q=1$ analog of the AW-function is the Wilson function, which is a non-polynomial generalization of the Wilson polynomials [17]. The second level consists of the big and little q-Jacobi functions, and both these functions arise as limits of the AW-function.

Our goal in this paper is to provide a complete hyperbolic non-polynomial q-Askey scheme. In fact, we will only be interested in continuous (as opposed to discrete) orthogonal polynomials and we will therefore only consider the part of the q-Askey scheme originating from the Askey–Wilson polynomials shown in Fig. 1. Just like the Askey and q-Askey schemes, the non-polynomial scheme we present has five levels, see Fig. 2. Whereas each element in the q-Askey scheme is a family of polynomials, each element of the non-polynomial scheme is a meromorphic function. These meromorphic functions are given by contour integrals whose integrands are built from Ruijsenaars’ hyperbolic gamma function [50] (alternatively, the integrands can be expressed in terms of Faddeev’s quantum dilogarithm [14], Woronowicz’s quantum exponential [62], or Kurokawa’s double sine function [37]). The non-polynomial scheme is a generalization of the q-Askey scheme in the sense that each element in Fig. 1 is obtained from an element at the same level in Fig. 2 when one of the variables is suitably discretized.

Just like in the q-Askey scheme, the elements in the non-polynomial scheme are joint eigenfunctions. Indeed, we will show that each meromorphic function in the non-polynomial scheme satisfies two pairs of difference equations. In the polynomial limit, one of these pairs reduces to the recurrence relation, while the other pair reduces to the q-difference equation.

The first (top) level of the non-polynomial scheme consists of a single function, namely, Ruijsenaars’ hypergeometric function $\mathcal {R}$¹ [51]. Below this top level, there are four further levels involving functions which can be obtained from $\mathcal {R}$ by taking various limits; we denote these functions by $\mathcal {H}$, $\mathcal {S}$, $\mathcal {X}$, $\mathcal {Q}$, $\mathcal {L}$, $\mathcal {W}$, and $\mathcal {M}$, where the letters are chosen so that $\mathcal {H}$ is a non-polynomial generalization of the continuous dual q-Hahn polynomials which are denoted by $H_n$ in “Appendix B”, $\mathcal {S}$ is a non-polynomial generalization of the Al-Salam Chihara polynomials which are denoted by $S_n$ in “Appendix B”, etc.

Fig. 1

The part of the q-Askey scheme relevant for the present paper

Fig. 2

The elements of the scheme constructed in the present paper. It is a non-polynomial generalization of the scheme in Fig. 1

Each of the functions in the non-polynomial scheme depends on a number of parameters as well as two variables. For example, $\mathcal {H}$ depends on the parameters $b, \theta _0, \theta _t, \theta _*$ as well as the two variables $\sigma _s, \nu$, while $\mathcal {S}$ depends on the parameters $b, \theta _0, \theta _t$ as well as the two variables $\sigma _s, \rho$, see Fig. 2. As a matter of notation, we will refer to the top level of the scheme (which involves $\mathcal {R}$) as the first level, to the next level (which involves $\mathcal {H}$) as the second level, etc. Each function at levels 2–5 will be defined as a limit of the function above it in the scheme. We will show that this limit exists (at least for a certain range of the two variables). We will also derive an integral representation for the function, show that it extends to a meromorphic function of each of the two variables everywhere in the complex plane, and establish two pairs of difference equations (one in each of the two variables). Finally, we will establish the polynomial limit to the corresponding element in the q-Askey scheme. Actually, two of the functions in the non-polynomial scheme, namely $\mathcal {H}$ and $\mathcal {S}$, possess two different polynomial limits: $\mathcal {H}$ reduces to the continuous dual q-Hahn polynomials when $\nu$ is discretized and to the big q-Jacobi polynomials when $\sigma _s$ is discretized; similarly, $\mathcal {S}$ reduces to the Al-Salam Chihara polynomials when $\rho$ is discretized and to the little q-Jacobi polynomials when $\sigma _s$ is discretized.

The orthogonal polynomials in the Askey and q-Askey schemes are of fundamental importance in a wide variety of fields. Elements in the q-Askey scheme have found applications, for instance, in models of statistical mechanics [11, 56, 60, 61], in representation theory of quantum algebras [1, 4, 15, 30, 31, 33, 34, 42, 43, 63], and in the geometry of Painlevé equations [41]. We expect the functions in the proposed non-polynomial scheme to also be relevant in many different contexts (see Sect. 1.1 for a related discussion). This is certainly true of the top element, Ruijsenaars’ $\mathcal {R}$-function. As an example of the broad relevance of $\mathcal {R}$, we note that one of the authors recently showed [48] that $\mathcal {R}$ is equivalent (up to a change of variables) to the Virasoro fusion kernel which is a central object in conformal field theory [10, 45,46,47]. In fact, it was in the context of conformal field theory that we first conceived of the non-polynomial scheme presented in this paper. First, in [38], we introduced a family of confluent Virasoro fusion kernels $\mathcal {C}_k$ while studying confluent conformal blocks of the second kind of the Virasoro algebra. Later, we realized that the $\mathcal {C}_k$ can be viewed as non-polynomial generalizations of the continuous dual q-Hahn and the big q-Jacobi polynomials, which led us to conjecture that there exists a non-polynomial generalization of the q-Askey scheme with the Virasoro fusion kernel as its top member [39]. In this paper, we prove this conjecture. However, instead of adopting the Virasoro fusion kernel as the top element of the scheme, we use Ruijsenaars’ $\mathcal {R}$-function as our starting point. The result of [48] implies that these two choices are equivalent, but we have found that the scheme originating from $\mathcal {R}$ is simpler and mathematically more convenient.

Let us also mention that a more general degeneration scheme of hyperbolic hypergeometric integrals was studied in [8, 9] (see in particular [8, Fig. 5.8]) whose top-family consists of a hyperbolic hypergeometric integral depending on 8 parameters and possessing an $E_7$-symmetry. It was shown in [9, Sect. 4.6] that Ruijsenaars’ $\mathcal R$-function is a special case of this function. Although [8] and [9] do not focus on constructing non-polynomial extensions of Askey schemes, the results of these works are relevant for the constructions in this paper. Another relevant work is [5] where a systematic description of limits from elliptic hypergeometric functions to basic hypergeometric functions is described.

1.1 Relations to Other Areas

We discuss some relations that deserve to be studied in more detail in the future.

1.1.1 Relation to Quantum Relativistic Integrable Systems

The function $\mathcal {R}$ was introduced in [49] in the context of relativistic systems of Calogero-Moser type, and studied in greater detail in [51,52,53]. In particular, after a proper change of parameters, the difference operator $D_\mathcal {R}$ defined in (3.6) corresponds to the rank one hyperbolic van Diejen Hamiltonian. Strong coupling (or Toda) limits of $D_\mathcal {R}$ and of its higher rank generalizations were considered in [12]. In this paper, we have shown that each element in the non-polynomial scheme is a joint eigenfunction of four difference operators, which, by construction, are confluent limits of $D_\mathcal {R}$. Thus it would be desirable to understand if each confluent limit considered in the present article corresponds to a Toda-type limit. In fact, a renormalized version of the function $\mathcal {Q}$ (see (7.6)), which is one of the two elements at the fifth level of the non-polynomial scheme, was studied in [24, 54] and was interpreted as the eigenfunction of a q-Toda type Hamiltonian.

1.1.2 Relation to Two-Dimensional CFTs

As mentioned in the introduction, Ruijsenaars’ hypergeometric function is essentially equal to the Virasoro fusion kernel [48]. The latter plays a key role in the conformal bootstrap approach to Liouville conformal field theory on punctured Riemann spheres [45]. The Virasoro fusion kernel also appears in four-dimensional supersymmetric gauge theory as a result of the AGT correspondence [2]. We believe that the various confluent limits considered in the present article correspond to collisions of punctures in Liouville theory. Therefore, we expect the other elements in the non-polynomial scheme to play a role in Liouville theory on different Riemann surfaces with punctures and cusps [16].

1.1.3 Relation to Quantum Groups and Double Affine Hecke Algebras (DAHA)

Families in the Askey scheme have a well-understood group theoretic interpretation [32]. As for the q-Askey scheme, the AW polynomials are well-rooted in the representation theory of quantum groups [43] and of DAHA [44]. Similar interpretations have been found for the AW-function [57, 58]. On the other hand, motivated by the application of quantum groups in two-dimensional conformal field theories, Faddeev introduced in [13] the modular double of $U_q(sl_2(\mathbb {R}))$, denoted $U_{q\tilde{q}}(sl_2(\mathbb {R}))$. Both the functions $\mathcal R$ and $\mathcal Q$ can be obtained from the viewpoint of harmonic analysis on $U_{q\tilde{q}}(sl_2(\mathbb {R}))$ [7, 24, 45, 46]. More generally, we expect that all the elements of the non-polynomial scheme can be obtained from the viewpoint of harmonic analysis on $U_{q\tilde{q}}(g)$, where g is the Lie algebra of a non-compact Lie group. Examples of such modular doubles were studied in [20,21,22]. Finally, an understanding of the $\mathcal R$-function from the viewpoint of DAHA is currently lacking.

1.2 Outlook

It was shown in [52, Theorem 1.1] that a renormalized version of the function $\mathcal {R}$ has a hidden $D_4$ symmetry in its four external parameters. It would be interesting to determine what this $D_4$ invariance becomes after taking the various confluent limits presented in this article.

The $\mathcal {R}$-function can be expressed as a sum of two terms, where each term is proportional to a product of two AW-functions [9, Theorem 6.5] (this is a non-polynomial analog of the hyperbolic extension of the q-Askey scheme described in the introduction). It is therefore natural to study the relation between our non-polynomial hyperbolic scheme and the non-polynomial trigonometric scheme of [27].

A Fourier transform admitting the AW-function as kernel was constructed in [28] (see also [26, 29] for the little and big q-Jacobi functions, respectively). Moreover, in a certain region of the external parameter space, Ruijsenaars constructed in [53] a unitary Hilbert space transform admitting the function $\mathcal R$ as a kernel. An important project is to handle the Hilbert space theory of the other difference equations studied in this article. This project would be of particular interest, since the theory of linear (analytic) difference equations is much less understood than the theory of linear discrete difference, or linear differential equations. In particular, it is not known if the existence of joint eigenfunctions implies that they can be promoted to kernels of unitary Hilbert space transforms.

Furthermore, the Askey–Wilson polynomials $R_n(z;\alpha ,\beta ,\gamma ,\delta ,q)$ are symmetric under the exchange $z \rightarrow z^{-1}$. The non-symmetric Askey–Wilson polynomials, introduced in [44, 55], are Laurent polynomials satisfying a difference/reflection equation. A symmetrization procedure described in [44] can be used to recover the symmetric Askey–Wilson polynomials from their non-symmetric counterparts. More generally, a non-symmetric extension of the q-Askey scheme was initiated in [36, 40]. Finally, a non-symmetric AW-function, as well as its associated Fourier transform, was constructed in [57]. It would be interesting to construct a non-symmetric $\mathcal R$-function generalizing the non-symmetric AW-function, and study its transition limits.

1.3 Organization of the Paper

In Sect. 2, we introduce the function $s_b(z)$ which is the basic building block used to define the elements of the non-polynomial scheme. The eight functions $\mathcal {R}, \mathcal {H}, \mathcal {S}, \mathcal {X}, \mathcal {Q}, \mathcal {L}, \mathcal {W}, \mathcal {M}$ that make up the non-polynomial scheme are considered one by one in the eight Sects. 3–10. In Sect. 11, we derive—as an easy application of the non-polynomial scheme—a few duality formulas that relate members of the q-Askey scheme. In the two appendices, we have collected relevant definitions and properties of q-hypergeometric series and of the q-Askey scheme.

1.4 Standing Assumption

Throughout the paper, we make the following assumption.

Assumption 1.1

(Restrictions on the parameters) We assume that

$$\begin{aligned} (b,\theta _\infty ,\theta _1,\theta _t,\theta _0,\theta _*,\theta ) \in (0,\infty ) \times \mathbb {R}^6.\end{aligned}$$

(1.1)

Assumption 1.1 is made primarily for simplicity; we expect that all our results admit meromorphic continuations to more general values of the parameters, such as $b\notin i\mathbb {R}$ and $(\theta _\infty ,\theta _1,\theta _t,\theta _0,\theta _*,\theta ) \in \mathbb {C}^6$.

1.5 Notational Conventions

Our choice of parameters for the $\mathcal R$-function has its origin in conformal field theory. The parameters $\theta _0,\theta _t,\theta _1,\theta _\infty$ represent the conformal dimensions of the four Virasoro primary fields. This notation is also frequently used in the context of the Painlevé VI equation [19]. Moreover, the parameters $\sigma _s$ and $\sigma _t$ are the internal momenta characterizing the Virasoro conformal blocks in the s- and t-channels, respectively [48]. Finally, the parameter b characterizing the quantum deformation parameters $q= e^{2i \pi b^2}$ and ${\tilde{q}} = e^{2i\pi b^{-2}}$ parametrizes the central charge in conformal field theory as $c=1+6(b+1/b)^2$.

2 The Function $s_b(z)$

The elements of the non-polynomial scheme presented in this article are given by contour integrals, whose integrands involve the function $s_b(z)$ defined by

$$\begin{aligned} s_b(z)={\text {exp}}{\left[ i \int _0^\infty \frac{dy}{y} \left( \frac{{\text {sin}}{2yz}}{2{\text {sinh}}(b^{-1}y){\text {sinh}}(by)}-\frac{z}{y}\right) \right] }, \qquad |\text {Im} \,z|<\frac{Q}{2}, \end{aligned}$$

(2.1)

where $Q:= b + b^{-1}$. The function $s_b(z)$ is closely related to Ruijsenaars’ hyperbolic gamma function [50], Faddeev’s quantum dilogarithm function [14], Woronowicz’s quantum exponential function [62], and Kurokawa’s double sine function [37]. More precisely, it is related to Ruijsenaars’ hyperbolic gamma function G in [51, Eq. (A.3)] by

$$\begin{aligned} s_b(z) = G(b, b^{-1}; z). \end{aligned}$$

(2.2)

It follows from (2.2) and the results in [51] that $s_b(z)$ is a meromorphic function of $z \in \mathbb {C}$ with zeros $\{z_{m,l}\}_{m,l=0}^\infty$ and poles $\{p_{m,l}\}_{m,l=0}^\infty$ located at

$$\begin{aligned} \begin{aligned}&z_{m,l}=\frac{i Q}{2}+i m b +il b^{-1}, \qquad m,l = 0,1, 2,\dots , \qquad (\text {zeros}), \\&p_{m,l}=-\frac{i Q}{2} -i m b-il b^{-1}, \qquad m,l = 0,1, 2,\dots , \qquad (\text {poles}). \end{aligned} \end{aligned}$$

(2.3)

The multiplicity of the zero $z_{m,l}$ in (2.3) is given by the number of distinct pairs $(m_i,l_i) \in \mathbb {Z}_{\ge 0} \times \mathbb {Z}_{\ge 0}$ such that $z_{m_i,l_i}=z_{m,l}$. The pole $p_{m,l}$ has the same multiplicity as the zero $z_{m,l}$. In particular, if $b^2$ is an irrational real number, then all the zeros and poles in (2.3) are distinct and simple. The residue $s_b$ at the simple pole $z=-iQ/2$ is given by

$$\begin{aligned} \underset{z=-\frac{iQ}{2}}{\text {Res}} s_b(z) =\frac{i}{2\pi }. \end{aligned}$$

(2.4)

Furthermore, $s_b$ is a meromorphic solution of the following pair of difference equations:

$$\begin{aligned} \frac{s_b(z+\frac{ib}{2})}{s_b(z-\frac{ib}{2})}=2{\text {cosh}}{\pi b z}, \qquad \frac{s_b(z+\frac{i}{2b})}{s_b(z-\frac{i}{2b})}=2{\text {cosh}}{\frac{\pi z}{b}}. \end{aligned}$$

(2.5)

Applying the difference equations (2.5) recursively, it can be verified that, for any integer $m \ge 0$,

$$\begin{aligned}&\frac{s_b(x+imb^{\pm 1})}{s_b(x)} = e^{-\frac{\pi b^{\pm 1} m}{2} (2 x+i b^{\pm 1} m)} \left( -e^{i \pi b^{\pm 2}} e^{2 \pi b^{\pm 1} x};e^{2 i \pi b^{\pm 2}}\right) _m, \end{aligned}$$

(2.6)

where the q-Pochhammer symbol $(a;q)_m$ is defined in “Appendix A”. Finally, the function $s_b(z)$ has the obvious symmetry

$$\begin{aligned} s_b(z) = s_{b^{-1}}(z) \end{aligned}$$

(2.7)

and possesses an asymptotic formula which is a consequence of [51, Theorem A.1] and (2.2): For each $\epsilon >0$,

$$\begin{aligned} \pm \ln s_b(z) = -\frac{i\pi z^2}{2} - \frac{i\pi }{24}(b^2 + b^{-2}) + O(e^{-\frac{2\pi (1-\epsilon )}{\max (b, b^{-1})}|\text {Re} \,z|}), \qquad \text {Re} \,z \rightarrow \pm \infty , \end{aligned}$$

(2.8)

uniformly for $(b,\text {Im} \,z)$ in compact subsets of $(0,\infty ) \times \mathbb {R}$.

3 The Function $\mathcal {R}$

The top element of the non-polynomial q-Askey scheme presented in this paper is Ruijsenaars’ hypergeometric function $\mathcal {R}$ [51]. Using the notation of [48], this function can be expressed as

$$\begin{aligned} \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array}; \begin{array}{c} \sigma _s \vspace{0.15cm} \\ \sigma _t \end{array}, b\right] = P_\mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \sigma _t \end{array},b\right] \int _{\mathcal {C}_{\mathcal {R}}} dx ~ I_\mathcal {R}\left[ x;\begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \sigma _t \end{array},b\right] ,\end{aligned}$$

(3.1)

where the prefactor $P_{\mathcal {R}}$ is given by

$$\begin{aligned}{} & {} P_\mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \sigma _t \end{array},b\right] = s_b\left( \tfrac{iQ}{2}+2\theta _t \right) \nonumber \\{} & {} \quad \times \prod _{\epsilon = \pm 1} s_b\left( \tfrac{iQ}{2}+\theta _0+\theta _1+\epsilon \theta _\infty +\theta _t \right) s_b(\epsilon \sigma _s-\theta _0-\theta _t)s_b(\epsilon \sigma _t-\theta _1-\theta _t),\nonumber \\ \end{aligned}$$

(3.2)

and the integrand $I_{\mathcal {R}}$ is defined by

$$\begin{aligned} I_\mathcal {R}\left[ x;\begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \sigma _t \end{array},b\right]{} & {} = \frac{1}{s_b(x+\tfrac{i Q}{2}) s_b(x+\tfrac{i Q}{2}+2 \theta _t)}\nonumber \\{} & {} \times \prod _{\epsilon =\pm 1}\frac{s_b(x+\theta _0+\theta _t+\epsilon \sigma _s) s_b(x+\theta _1+\theta _t+\epsilon \sigma _t)}{s_b\left( x+\tfrac{i Q}{2}+\theta _0+\theta _1+\epsilon \theta _\infty +\theta _t\right) }.\nonumber \\ \end{aligned}$$

(3.3)

In view of (2.3), the integrand $I_\mathcal {R}$ possesses eight semi-infinite sequences of poles in the complex x-plane. With the restriction that $b>0$ imposed by Assumption 1.1, there are four vertical downward sequences starting at $x =- \theta _0-\theta _t\pm \sigma _s -\frac{i Q}{2}$ and $x = - \theta _1 - \theta _t \pm \sigma _t -\frac{i Q}{2}$, and four vertical upward sequences starting at $x=0$, $x=-2\theta _t$, and $x=-\theta _0-\theta _1\pm \theta _\infty -\theta _t$. The contour $\mathcal {C}_{\mathcal {R}}$ in (3.1) is any curve from $-\infty$ to $+\infty$ which separates the four upward from the four downward sequences of poles. If in addition to Assumption 1.1, we also assume that $\sigma _s, \sigma _t \in \mathbb {R}$, then the contour of integration $\mathcal {C}_{\mathcal {R}}$ can be chosen to be any curve from $-\infty$ to $+\infty$ lying in the open strip $\text {Im} \,x\in (-Q/2,0)$.

Thanks to the symmetry (2.7) of the function $s_b$, we have

$$\begin{aligned} \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \sigma _t \end{array},b^{-1}\right] = \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \sigma _t \end{array},b\right] . \end{aligned}$$

(3.4)

Moreover, it follows directly from (3.1) that $\mathcal {R}$ is even in each of the variables $\sigma _s$ and $\sigma _t$, and that it satisfies the following self-duality symmetry:

$$\begin{aligned} \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \sigma _t \end{array},b\right] = \mathcal {R}\left[ \begin{array}{c} \theta _0\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _1 \end{array};\begin{array}{c} \sigma _t \vspace{0.15cm} \\ \sigma _s \end{array},b\right] . \end{aligned}$$

(3.5)

3.1 Difference Equations

Let $e^{\pm i b \partial _x}$ be the translation operator which formally acts on meromorphic functions f(x) by $e^{\pm i b \partial _x} f(x)=f(x\pm i b)$. Let $D_\mathcal {R}$ be the difference operator defined by

$$\begin{aligned} D_\mathcal {R} \left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} b,\;\sigma _s \end{array}\right]= & {} d_\mathcal {R}^+ \left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} b,\;\sigma _s \end{array}\right] e^{ib \partial _{\sigma _s}} + d_\mathcal {R}^+ \left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} b,\;-\sigma _s \end{array}\right] e^{-ib \partial _{\sigma _s}} \nonumber \\{} & {} \quad +\, d_\mathcal {R}^0 \left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} b,\;\sigma _s \end{array}\right] , \end{aligned}$$

(3.6)

where

$$\begin{aligned}{} & {} d_\mathcal {R}^+ \left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} b,\;\sigma _s \end{array}\right] \nonumber \\{} & {} \quad = - \frac{4 \prod _{\epsilon = \pm 1} \cosh \left( \pi b \left( \frac{ib}{2} +\theta _t+\sigma _s+\epsilon \theta _0\right) \right) \cosh \left( \pi b \left( \frac{ib}{2}+\theta _1+\sigma _s+\epsilon \theta _\infty \right) \right) }{\sinh (2 \pi b \sigma _s) \sinh (\pi b (2 \sigma _s+i b))}\nonumber \\ \end{aligned}$$

(3.7)

and

$$\begin{aligned} \begin{aligned}&d_\mathcal {R}^0\left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} b,\;\sigma _s \end{array}\right] \\&\quad = -2\cosh {( 2\pi b (\theta _1+\theta _t+\tfrac{ib}{2}))} - d_\mathcal {R}^+ \left[ \begin{array}{c} \theta _1\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} b,\;\sigma _s \end{array}\right] - d_\mathcal {R}^+ \left[ \begin{array}{c} \theta _1\;\;\theta _t\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} b,\;-\sigma _s \end{array}\right] . \end{aligned} \end{aligned}$$

(3.8)

For $(\sigma _s,\sigma _t) \in \mathbb C^2$, the function $\mathcal {R}$ satisfies the following four difference equations [51] (using the notation of [48]):

$$\begin{aligned}{} & {} D_\mathcal {R} \left[ \begin{array}{c} \theta _1\;\;\theta _t\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} b,\;\sigma _s \end{array}\right] \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \\ \sigma _t \end{array},b\right] = 2\cosh {(2\pi b \sigma _t)} \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \\ \sigma _t \end{array},b\right] ,\qquad \end{aligned}$$

(3.9a)

$$\begin{aligned}{} & {} D_\mathcal {R} \left[ \begin{array}{c} \theta _1\;\;\theta _t\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} b^{-1},\;\sigma _s \end{array}\right] \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \\ \sigma _t \end{array},b\right] = 2\cosh {(2\pi b^{-1} \sigma _t)} \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t\\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \\ \sigma _t \end{array},b\right] ,\qquad \end{aligned}$$

(3.9b)

$$\begin{aligned}{} & {} D_\mathcal {R} \left[ \begin{array}{c} \theta _0\;\;\theta _t \\ \theta _{\infty }\;\theta _1 \end{array};\begin{array}{c} b,\;\sigma _t \end{array}\right] \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t \\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \\ \sigma _t \end{array},b\right] = 2\cosh {(2\pi b \sigma _s)} \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t \\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \\ \sigma _t \end{array},b\right] ,\qquad \end{aligned}$$

(3.9c)

$$\begin{aligned}{} & {} D_\mathcal {R} \left[ \begin{array}{c} \theta _0\;\;\theta _t \\ \theta _{\infty }\;\theta _1 \end{array};\begin{array}{c} b^{-1},\;\sigma _t \end{array}\right] \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t \\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \\ \sigma _t \end{array},b\right] = 2\cosh {(2\pi b^{-1} \sigma _s)} \mathcal {R}\left[ \begin{array}{c} \theta _1\;\;\theta _t \\ \theta _{\infty }\;\theta _0 \end{array};\begin{array}{c} \sigma _s \\ \sigma _t \end{array},b\right] .\qquad \end{aligned}$$

(3.9d)

Note that the difference equations (3.9b), (3.9c), and (3.9d) follow from (3.9a) and the symmetries (3.4)–(3.5) of the function $\mathcal {R}$.

3.2 Polynomial Limit

It was shown in [51] that the function $\mathcal {R}$ reduces to the Askey–Wilson polynomials when one of the variables $\sigma _s$ and $\sigma _t$ is suitably discretized. This result was reobtained in the CFT setting in [39]. We now recall the result of [39]. In addition to Assumption 1.1, we need the following assumption.

Assumption 3.1

(Restriction on the parameters) Assume that $b > 0$ is such that $b^2$ is irrational, and that, for $\epsilon ,\epsilon ' = \pm 1,$

$$\begin{aligned} \begin{aligned}&\theta _\infty , \theta _t, \text {Re} \,\sigma _s, \text {Re} \,\sigma _t \ne 0, \qquad \text {Re} \,\big (\theta _0 - \theta _1 + \epsilon \sigma _s + \epsilon ' \sigma _t\big ) \ne 0, \\&\quad \theta _0+\theta _1+\epsilon \theta _\infty +\epsilon '\theta _t \ne 0. \end{aligned}\end{aligned}$$

(3.10)

Assumption 3.1 implies that the four increasing and the four decreasing sequences of poles of the integrand in (3.3) do not overlap. The assumption that $b^2$ is irrational implies that all the poles of the integrand are simple.

Theorem 3.2

[39, Theorem 4.2] Suppose Assumptions 1.1 and 3.1 are satisfied. Define $\{\sigma _s^{(n)}\}_{n=0}^\infty \subset \mathbb {C}$ by

$$\begin{aligned} \sigma _s^{(n)} = \theta _0+\theta _t+\tfrac{iQ}{2}+ibn. \end{aligned}$$

(3.11)

Under the parameter correspondence

$$\begin{aligned}{} & {} \alpha _R =e^{2 \pi b \left( \frac{i Q}{2}+\theta _1+\theta _t\right) }, \quad \beta _R =e^{2 \pi b \left( \frac{i Q}{2}+\theta _0-\theta _\infty \right) }, \quad \gamma _R=e^{2 \pi b \left( \frac{i Q}{2}-\theta _1+\theta _t\right) }, \nonumber \\{} & {} \quad \delta _R=e^{2 \pi b \left( \frac{i Q}{2}+\theta _0+\theta _\infty \right) }, q=e^{2i\pi b^2}, \end{aligned}$$

(3.12)

the Ruijsenaars hypergeometric function $\mathcal {R}$ defined in (3.1) satisfies, for each integer $n \ge 0$,

$$\begin{aligned} \lim \limits _{\sigma _s \rightarrow \sigma _s^{(n)}} \mathcal {R} \left[ \begin{array}{c} \theta _1\;\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\;\theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \sigma _t \end{array},b\right] = R_n(e^{2\pi b \sigma _t};\alpha _R,\beta _R,\gamma _R,\delta _R,q), \end{aligned}$$

(3.13)

where $R_n$ are the Askey–Wilson polynomials defined in (B.1).

Remark 3.3

[39, Remark 4.3] The result of Theorem 3.2 can be generalized as follows. Instead of considering the limit of $\mathcal R$ as $\sigma _s$ approaches one of the points $\sigma _s^{(n)}$ defined in (3.11), we can consider the limit

$$\begin{aligned} \sigma _s \rightarrow \sigma _s^{(n,m)}:= \sigma _s^{(n)} + \frac{im}{b}, \end{aligned}$$

(3.14)

for any integers $n,m \ge 0$. In this limit, $\mathcal R$ reduces to a product of two Askey–Wilson polynomials of the form $R_n \times R_m$. The first polynomial $R_n$ is expressed in terms of the quantum deformation parameter $q=e^{2i\pi b^2}$, while the second polynomial $R_m$ is expressed in terms of $\tilde{q}=e^{2i\pi b^{-2}}$. In the case $m = 0$ treated in Theorem 3.2, the second polynomial reduces to $R_0 = 1$.

A generalization similar to the one described in Remark 3.3 can be formulated for each of the families described in this article, thereby providing a limit from our scheme to the hyperbolic extension of the q-Askey scheme. As a matter of brevity, the details of this construction will be omitted.

4 The Function $\mathcal {H}$

The Askey–Wilson polynomials $R_n$ form the top element of the q-Askey scheme. Just like the elements of the q-Askey scheme are obtained from the polynomials $R_n$ via various limiting procedures, the elements of the non-polynomial scheme we present in this paper are obtained by taking various limits of Ruijsenaars’ hypergeometric function $\mathcal {R}$. The second level of the non-polynomial scheme involves the function $\mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu )$, which is defined as the confluent limit $\Lambda \rightarrow -\infty$ of the top element $\mathcal {R}$ evaluated at

$$\begin{aligned} \theta _\infty = \frac{\Lambda -\theta _*}{2}, \qquad \theta _1 = \frac{\Lambda +\theta _*}{2}, \qquad \sigma _t = \frac{\Lambda }{2} + \nu . \end{aligned}$$

(4.1)

In this section, we derive an integral representation for the function $\mathcal {H}$ and we show that it is a joint eigenfunction of four difference operators, two acting on $\sigma _s$ and the other two on $\nu$. We also show that $\mathcal {H}$ reduces to the continuous dual q-Hahn and the big q-Jacobi polynomials when $\nu$ and $\sigma _s$ are suitably discretized, respectively. Since these polynomials lie at the second level of the q-Askey scheme, this shows that $\mathcal {H}$ indeed provides a natural non-polynomial generalization of the elements at the second level.

4.1 Definition and Integral Representation

Definition 4.1

The function $\mathcal {H}$ is defined by

$$\begin{aligned} \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = \lim \limits _{\Lambda \rightarrow -\infty } \mathcal {R}\left[ \begin{array}{c} \frac{\Lambda +\theta _*}{2}\;\; \theta _t\vspace{0.1cm}\\ \frac{\Lambda -\theta _*}{2}\;\; \theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \frac{\Lambda }{2}+\nu \end{array},b\right] . \end{aligned}$$

(4.2)

The next theorem shows that, for each choice of $(b,\theta _0,\theta _t,\theta _*) \in (0,\infty ) \times \mathbb {R}^3$, $\mathcal {H}$ is a well-defined and analytic function of

$$\begin{aligned} (\sigma _s,\nu ) \in (\mathbb {C} \setminus \Delta _{\mathcal {H}, \sigma _s}) \times \big (\{\text {Im} \,\nu > -Q/2\} \setminus \Delta _{\nu }\big ), \end{aligned}$$

(4.3)

where $\Delta _{\mathcal {H}, \sigma _s}, \Delta _{\nu } \subset \mathbb {C}$ are discrete sets of points at which $\mathcal {H}$ may have poles. In particular, $\mathcal {H}$ is a meromorphic function of $\sigma _s \in \mathbb {C}$ and of $\nu$ for $\text {Im} \,\nu > -Q/2$. More precisely, $\Delta _{\mathcal {H}, \sigma _s}$ and $\Delta _{\nu }$ are given by

$$\begin{aligned} \Delta _{\mathcal {H}, \sigma _s} :=&\; \{\pm \sigma _s \, |\, \sigma _s \in \Delta _{\mathcal {H}, \sigma _s}'\}, \\ \Delta _{\nu } :=&\; \{\tfrac{\theta _*}{2} \pm \theta _t + \tfrac{iQ}{2} + i m b +il b^{-1}\}_{m,l=0}^\infty \cup \{-\tfrac{\theta _*}{2} - \theta _0 + \tfrac{iQ}{2} + i m b +il b^{-1}\}_{m,l=0}^\infty \nonumber \end{aligned}$$

(4.4a)

$$\begin{aligned}&\cup \{\tfrac{\theta _*}{2}+ \theta _t -\tfrac{i Q}{2} -i m b-il b^{-1}\}_{m,l=0}^\infty \end{aligned}$$

(4.4b)

where

$$\begin{aligned} \Delta _{\mathcal {H}, \sigma _s}' :=&\; \{\theta _0 \pm \theta _t + \tfrac{iQ}{2} + i m b +il b^{-1}\}_{m,l=0}^\infty \cup \{-\theta _* + \tfrac{iQ}{2} + i m b +il b^{-1}\}_{m,l=0}^\infty \\&\cup \{\theta _0+ \theta _t -\tfrac{i Q}{2} -i m b-il b^{-1}\}_{m,l=0}^\infty . \end{aligned}$$

The theorem also provides an integral representation for $\mathcal {H}$ for $(\sigma _s,\nu )$ satisfying (4.3). In fact, even if the requirement $\text {Im} \,\nu > -Q/2$ is needed to ensure convergence of the integral in the integral representation for $\mathcal {H}$, we will show later in this section, with the help of the difference equations satisfied by $\mathcal {H}$, that $\mathcal {H}$ extends to a meromorphic function of $(\sigma _s, \nu ) \in \mathbb {C}^2$.

Theorem 4.2

Suppose that Assumption 1.1 holds. Let $\Delta _\mathcal {H,\sigma _s}, \Delta _{\nu } \subset \mathbb {C}$ be the discrete subsets defined in (4.4). Then the limit in (4.2) exists uniformly for $(\sigma _s,\nu )$ in compact subsets of

$$\begin{aligned} \Omega _{\mathcal {H}}:= (\mathbb {C} \setminus \Delta _{\mathcal H,\sigma _s}) \times \big (\{\text {Im} \,\nu > -Q/2\} \setminus \Delta _{\nu }\big ). \end{aligned}$$

Moreover, $\mathcal {H}$ is an analytic function of $(\sigma _s,\nu ) \in \Omega _{\mathcal {H}}$ and admits the following integral representation:

$$\begin{aligned} \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = P_\mathcal {H}\left( \sigma _s,\nu \right) \displaystyle \int _{\mathcal {C}_{\mathcal {H}}} dx ~ I_\mathcal {H}\left( x, \sigma _s,\nu \right) \qquad \text {for }(\sigma _s,\nu ) \in \Omega _{\mathcal {H}},\end{aligned}$$

(4.5)

where the dependence of $P_\mathcal {H}$ and $I_\mathcal {H}$ on $b,\theta _0,\theta _t,\theta _*$ is omitted for simplicity,

$$\begin{aligned} P_\mathcal {H}\left( \sigma _s,\nu \right)&= s_b\left( 2 \theta _t+\tfrac{i Q}{2}\right) s_b\left( \theta _0+\theta _*+\theta _t+\tfrac{i Q}{2}\right) s_b\left( \nu -\tfrac{\theta _*}{2}-\theta _t \right) \nonumber \\&\quad \times \prod _{\epsilon = \pm 1} s_b\left( \epsilon \sigma _s-\theta _0-\theta _t\right) , \end{aligned}$$

(4.6)

$$\begin{aligned} I_\mathcal {H}\left( x,\sigma _s,\nu \right)&= e^{i \pi x \left( \frac{\theta _*}{2}-\theta _0+\nu -\frac{i Q}{2}\right) } \nonumber \\&\quad \,\times \frac{s_b\left( x+\frac{\theta _*}{2}+\theta _t-\nu \right) \prod _{\epsilon =\pm 1} s_b(x+\theta _0+\theta _t+\epsilon \sigma _s)}{s_b\left( x+\frac{i Q}{2}\right) s_b\left( x+2\theta _t+\frac{i Q}{2}\right) s_b\left( x+\theta _0+\theta _*+\theta _t+\frac{i Q}{2}\right) }, \end{aligned}$$

(4.7)

and the contour $\mathcal {C}_{\mathcal {H}}$ is any curve from $-\infty$ to $+\infty$ which separates the three upward from the three downward sequences of poles. In particular, $\mathcal {H}$ is a meromorphic function of $(\sigma _s, \nu ) \in \mathbb {C}\times \{\text {Im} \,\nu > -Q/2\}$. If $(\sigma _s,\nu ) \in \mathbb {R}^2$, then the contour $\mathcal {C}_{\mathcal {H}}$ can be any curve from $-\infty$ to $+\infty$ lying within the strip $\text {Im} \,{x} \in (-Q/2,0)$.

Proof

Let $(b,\theta _0,\theta _t,\theta _*) \in (0,\infty ) \times \mathbb {R}^3$. Using the identity $s_b(z)= 1/s_b(-z)$, it is straightforward to verify that

$$\begin{aligned}&P_\mathcal {R}\left[ \begin{array}{c} \frac{\Lambda +\theta _*}{2}\;\; \theta _t\vspace{0.1cm}\\ \frac{\Lambda -\theta _*}{2}\;\; \theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \frac{\Lambda }{2}+\nu \end{array}\;,b\right] I_\mathcal {R}\left[ x; \begin{array}{c} \frac{\Lambda +\theta _*}{2}\;\; \theta _t\vspace{0.1cm}\\ \frac{\Lambda -\theta _*}{2}\;\; \theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \frac{\Lambda }{2}+\nu \end{array} \;,b\right] = \; P_\mathcal {H}\left( \sigma _s,\nu \right) X(x,\Lambda ) I_\mathcal {H}\left( x, \sigma _s,\nu \right) , \end{aligned}$$

(4.8)

where the dependence of $X(x,\Lambda )$ on $b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu$ is omitted for simplicity, and

$$\begin{aligned}{} & {} X(x,\Lambda ) \nonumber \\{} & {} \quad = e^{i\pi x (\frac{iQ}{2}+\theta _0-\frac{\theta _*}{2}-\nu )} \frac{s_b\left( \Lambda +\theta _0+\theta _t +\frac{i Q}{2}\right) }{s_b\left( \Lambda +\frac{\theta _*}{2}+\theta _t+\nu \right) } \frac{s_b\left( x+\Lambda +\frac{\theta _*}{2}+\theta _t+\nu \right) }{s_b\left( x+\Lambda +\theta _0+\theta _t +\frac{i Q}{2}\right) }.\qquad \quad \end{aligned}$$

(4.9)

Due to the properties (2.3) of the function $s_b$, the function $I_\mathcal {H}(\cdot , \sigma _s, \nu )$ possesses three increasing sequences of poles starting at $x=0, x=-2\theta _t$ and $x =-\theta _0-\theta _*-\theta _t$, as well as three decreasing sequences of poles starting at $x= -\tfrac{iQ}{2}-\tfrac{\theta _*}{2}-\theta _t+\nu$ and $x=-\tfrac{iQ}{2} \pm \sigma _s-\theta _0-\theta _t$. The discrete sets $\Delta _{\mathcal H,\sigma _s}$ and $\Delta _{\nu }$ contain all the values of $\sigma _s$ and $\nu$, respectively, for which poles in any of the three increasing collide with poles in any of the decreasing sequences. Indeed, consider for example the decreasing sequence starting at $x= -\tfrac{iQ}{2}-\tfrac{\theta _*}{2}-\theta _t+\nu$ and the increasing sequence starting at $x=0$. Poles from these two sequences collide if and only if

$$\begin{aligned} \nu \in \{\tfrac{\theta _*}{2} + \theta _t +\tfrac{iQ}{2} + i m b +il b^{-1}\}_{m, l = 0}^\infty , \end{aligned}$$

giving rise to the first set on the right-hand side of (4.4b).

Similarly, (2.3) implies that $X(\cdot , \Lambda )$ possesses one increasing sequence of poles starting at $x = -\Lambda - \theta _0 - \theta _t$ and one decreasing sequence of poles starting at $x = -\tfrac{iQ}{2} -\Lambda - \tfrac{\theta _*}{2} - \theta _t - \nu$. The real parts of the poles in these two sequences tend to $+\infty$ as $\Lambda \rightarrow -\infty$. The increasing sequence lies in the half-plane $\text {Im} \,x \ge 0$ and the decreasing sequence lies in the half-plane $\text {Im} \,x \le -\text {Im} \,\nu - Q/2$.

The sets $\Delta _\mathcal {H,\sigma _s}$ and $\Delta _{\nu }$ also contain all the values of $\sigma _s$ and $\nu$ at which the prefactor $P_\mathcal {H}(\sigma _s,\nu )$ has poles. For example, $P_\mathcal {H}$ has poles originating from the factor $s_b\left( \nu -\tfrac{\theta _*}{2}-\theta _t \right)$ at

$$\begin{aligned} \nu = \tfrac{\theta _*}{2}+ \theta _t -\frac{i Q}{2} -i m b-il b^{-1}, \qquad m, l = 0,1, 2, \dots , \end{aligned}$$

giving rise to the last set on the right-hand side of (4.4b).

Let $K_{\sigma _s}$ be a compact subset of $\mathbb {C} {\setminus } \Delta _\mathcal {H,\sigma _s}$ and let $K_{\nu }$ be a compact subset of $\{\text {Im} \,\nu > -Q/2\} \setminus \Delta _{\nu }$. Suppose $(\sigma _s, \nu ) \in K_{\sigma _s} \times K_{\nu }$. Then, the above discussion shows that it is possible to choose a contour $\mathcal {C}_{\mathcal {H}} = \mathcal {C}_{\mathcal {H}}(\sigma _s, \nu )$ from $-\infty$ to $+\infty$ which separates the four upward from the four downward sequences of poles of $X(\cdot ,\Lambda ) I_\mathcal {H}\left( \cdot , \sigma _s,\nu \right)$. It also follows that if we let the right tail of $\mathcal {C}_{\mathcal {H}}$ approach the horizontal line $\text {Im} \,x = -\epsilon$ as $\text {Re} \,x \rightarrow +\infty$, where $\epsilon > 0$ is sufficiently small, then there exists a $N < 0$ such that $\mathcal {C}_{\mathcal {H}}$ can be chosen to be independent of $\Lambda$ for $\Lambda < -N$. Thus, for such a choice of $\mathcal {C}_{\mathcal {H}}$, (3.1) and (4.8) imply that, for all $(\sigma _s, \nu ) \in K_{\sigma _s} \times K_{\nu }$ and all $\Lambda < -N$,

$$\begin{aligned} \mathcal {R}\left[ \begin{array}{c} \frac{\Lambda +\theta _*}{2}\;\; \theta _t\vspace{0.1cm}\\ \frac{\Lambda -\theta _*}{2}\;\; \theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \frac{\Lambda }{2}+\nu \end{array},b\right] = P_\mathcal {H}\left( \sigma _s,\nu \right) \int _{\mathcal {C}_{\mathcal {H}}} dx X(x,\Lambda ) I_\mathcal {H}\left( x, \sigma _s,\nu \right) . \end{aligned}$$

(4.10)

If $(\sigma _s,\nu ) \in \mathbb {R}^2$, then $\mathcal {C}_{\mathcal {H}}$ can be any curve from $-\infty$ to $+\infty$ lying within the strip $\text {Im} \,{x} \in (-Q/2,0)$.

Utilizing the asymptotic formula (2.8) for the function $s_b$ with $\epsilon = 1/2$, we find that

$$\begin{aligned} {\text {ln}}{\left( X(x, \Lambda )\right) } = O\Big ( e^{-\frac{\pi |\Lambda |}{\max (b,b^{-1})} } \Big ), \qquad \Lambda \rightarrow -\infty , \end{aligned}$$

(4.11)

uniformly for $(\sigma _s, \nu ) \in K_{\sigma _s} \times K_{\nu }$ and for x in bounded subsets of $\mathcal {C}_{\mathcal {H}}$. We deduce that

$$\begin{aligned} \lim \limits _{\Lambda \rightarrow -\infty } X(x,\Lambda ) = 1, \end{aligned}$$

(4.12)

uniformly for $(\sigma _s, \nu ) \in K_{\sigma _s} \times K_{\nu }$ and x in bounded subsets of $\mathcal {C}_{\mathcal {H}}$.

Using the asymptotic formula (2.8) for $s_b$ with $\epsilon =1/2$, we find that $I_\mathcal {H}$ obeys the estimate

$$\begin{aligned} I_\mathcal {H}\left( x,\sigma _s,\nu \right) = {\left\{ \begin{array}{ll} O\left( e^{-2\pi (\frac{Q}{2} + \text {Im} \,\nu ) |\text {Re} \,x|} \right) , &{} \text {Re} \,x \rightarrow + \infty ,\\ O\left( e^{-2\pi Q |\text {Re} \,x|} \right) , &{} \text {Re} \,x \rightarrow - \infty , \end{array}\right. } \end{aligned}$$

(4.13)

uniformly for $(\sigma _s,\nu ) \in K_{\sigma _s} \times K_{\nu }$ and $\text {Im} \,x$ in compact subsets of $\mathbb {R}$. Since the contour $\mathcal {C}_{\mathcal {H}}$ stays a bounded distance away from the increasing and the decreasing pole sequences, we infer that there exists a constant $C_1>0$ such that

$$\begin{aligned} |I_\mathcal {H}\left( x,\sigma _s,\nu \right) |\le {\left\{ \begin{array}{ll} C_1e^{-2\pi (\frac{Q}{2} + \text {Im} \,\nu ) |\text {Re} \,x|}, &{} x \in \mathcal {C}_{\mathcal {H}}, ~ \text {Re} \,x \ge 0, \\ C_1e^{-2\pi Q |\text {Re} \,x|}, &{} x \in \mathcal {C}_{\mathcal {H}}, ~ \text {Re} \,x \le 0, \end{array}\right. } \end{aligned}$$

(4.14)

uniformly for $(\sigma _s,\nu ) \in K_{\sigma _s} \times K_{\nu }$. In particular, since $K_{\nu } \subset \{\text {Im} \,\nu > -Q/2\}$, $I_\mathcal {H}$ has exponential decay on the left and right tails of the contour $\mathcal {C}_{\mathcal {H}}$.

Suppose we can show that there exist constants $c>0$ and $C>0$ such that

$$\begin{aligned} |X(x,\Lambda ) I_\mathcal {H}\left( x, \sigma _s,\nu \right) | \le C e^{-c |\text {Re} \,x |} \end{aligned}$$

(4.15)

uniformly for all $\Lambda < -N$, $x \in \mathcal {C}_{\mathcal {H}}$, and $(\sigma _s, \nu ) \in K_{\sigma _s} \times K_{\nu }$. Then it follows from (4.10), (4.12), and Lebesgue’s dominated convergence theorem that the limit in (4.2) exists uniformly for $(\sigma _s, \nu ) \in K_{\sigma _s} \times K_{\nu }$ and is given by (4.5). Since $K_{\sigma _s} \subset \mathbb {C} {\setminus } \Delta _\mathcal {H,\sigma _s}$ and $K_{\nu } \subset \mathbb {C} {\setminus } \Delta _{\nu }$ are arbitrary compact subsets, this proves that the limit in (4.2) exists uniformly for $(\sigma _s,\nu )$ in compact subsets of $\Omega _{\mathcal {H}}$ and proves (4.5). Moreover, the analyticity of $\mathcal {H}$ as a function of $(\sigma _s,\nu ) \in \Omega _{\mathcal {H}}$ follows from the analyticity of $\mathcal {R}$ together with the uniform convergence on compact subsets. Alternatively, the analyticity of $\mathcal {H}$ in $\Omega _{\mathcal {H}}$ can be inferred directly from the representation (4.5). Indeed, the possible poles of $\mathcal {H}$ lie at such values of $(\sigma _s,\nu )$ at which either the prefactor $P_{\mathcal {H}}$ has a pole or at which the contour of integration gets pinched between two poles of the integrand $I_\mathcal {H}$, and the definitions of $\Delta _{\mathcal H,\sigma _s}$ and $\Delta _\nu$ exclude both of these situations. Thus to complete the proof of the theorem, it only remains to prove (4.15).

To prove (4.15), we need to estimate the function X defined in (4.9). The asymptotic formula (2.8) for $s_b$ with $\epsilon =1/2$ implies that there exist constants $C_2, C_3, C_4>0$ such that the inequalities

$$\begin{aligned}&\bigg |\frac{s_b\left( \Lambda +\theta _0+\theta _t +\frac{i Q}{2}\right) }{s_b\left( \Lambda +\frac{\theta _*}{2}+\theta _t+\nu \right) }\bigg | \le C_2 e^{\pi (\frac{Q }{2} - \text {Im} \,\nu ) |\Lambda |}, \qquad \Lambda < -N, \end{aligned}$$

(4.16)

$$\begin{aligned}&\bigg |\frac{s_b\left( x+\Lambda +\frac{\theta _*}{2}+\theta _t+\nu \right) }{s_b\left( x+\Lambda +\theta _0+\theta _t +\frac{i Q}{2}\right) } \bigg | \le C_3 e^{-\pi (\frac{Q}{2} - \text {Im} \,\nu )|\Lambda + \text {Re} \,{x}|}, \qquad x \in \mathcal {C}_{\mathcal {H}}, ~ \Lambda \in \mathbb {R}, \end{aligned}$$

(4.17)

$$\begin{aligned}&|e^{i\pi x (\frac{iQ}{2}+\theta _0-\frac{\theta _*}{2}-\nu )}| \le C_4 e^{-\pi (\frac{Q}{2}-\text {Im} \,\nu )\text {Re} \,{x}}, \qquad x \in \mathcal {C}_{\mathcal {H}}, \end{aligned}$$

(4.18)

hold uniformly for $(\sigma _s,\nu ) \in K_{\sigma _s} \times K_{\nu }$. Combining the above estimates, we infer that there exists a constant $C_5$ such that

$$\begin{aligned} |X(x,\Lambda )| \le C_5 e^{\pi (\frac{Q}{2}-\text {Im} \,\nu )(|\Lambda | - |\Lambda + \text {Re} \,{x}| - \text {Re} \,{x})}, \qquad x \in \mathcal {C}_{\mathcal {H}}, ~ \Lambda < -N, \end{aligned}$$

(4.19)

uniformly for $(\sigma _s,\nu ) \in K_{\sigma _s} \times K_{\nu }$. The inequality (4.19) can be rewritten as follows:

$$\begin{aligned} |X(x,\Lambda )| \le {\left\{ \begin{array}{ll} C_5 e^{-2\pi (\frac{Q}{2} - \text {Im} \,\nu )(\Lambda + \text {Re} \,{x})}, &{} \Lambda + \text {Re} \,{x} \ge 0, \\ C_5, &{} \Lambda + \text {Re} \,{x} \le 0, \end{array}\right. } \quad x \in \mathcal {C}_{\mathcal {H}}, ~ \Lambda <-N,\nonumber \\ \end{aligned}$$

(4.20)

uniformly for $(\sigma _s,\nu ) \in K_{\sigma _s} \times K_{\nu }$.

If $\nu \in K_{\nu }$ is such that $-Q/2 < \text {Im} \,\nu \le Q/2$, then (4.20) implies that |X| is uniformly bounded for all $x \in \mathcal {C}_{\mathcal {H}}$, $\Lambda <-N$, and $(\sigma _s,\nu ) \in K_{\sigma _s} \times K_{\nu }$; hence (4.15) follows from (4.14) in this case. On the other hand, if $\nu \in K_{\nu }$ is such that $\text {Im} \,\nu \ge Q/2$, then (4.14) and (4.20) yield the existence of a constant $C_6> 0$ independent of $x \in \mathcal {C}_{\mathcal {H}}$, $\Lambda < -N$, and $(\sigma _s,\nu ) \in K_{\sigma _s} \times K_{\nu }$ such that

$$\begin{aligned}{} & {} |X(x,\Lambda ) I_\mathcal {H}\left( x, \sigma _s,\nu \right) |\nonumber \\{} & {} \quad \le {\left\{ \begin{array}{ll} C_6e^{2\pi (\frac{Q}{2} - \text {Im} \,\nu )|\Lambda |}e^{-2\pi Q|\text {Re} \,{x}|} \le C_6e^{-2\pi Q|\text {Re} \,{x}|}, &{} \text {Re} \,{x} \ge -\Lambda , \\ C_6e^{-2\pi (\frac{Q}{2} + \text {Im} \,\nu ) |\text {Re} \,x|} \le C_6 e^{-2\pi Q|\text {Re} \,x|}, &{} 0 \le \text {Re} \,x \le -\Lambda , \\ C_6e^{-2\pi Q |\text {Re} \,x|}, &{} \text {Re} \,x \le 0, \end{array}\right. }\qquad \end{aligned}$$

(4.21)

which shows (4.15) also in this case. This completes the proof. $\square$

Furthermore, thanks to the symmetry (2.7) of $s_b$, we have

$$\begin{aligned} \mathcal {H}(b^{-1},\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ). \end{aligned}$$

(4.22)

4.2 Difference Equations

We now show that the four difference equations (3.9) satisfied by the function $\mathcal {R}$ survive in the confluent limit (4.2). This implies that the function $\mathcal {H}$ is a joint eigenfunction of four difference operators, two acting on $\sigma _s$ and the remaining two on $\nu$.

We know from Theorem 4.2 that $\mathcal {H}$ is a well-defined meromorphic function of $(\sigma _s, \nu ) \in \mathbb {C} \times \{\text {Im} \,\nu > -Q/2\}$. The difference equations will first be derived as equalities between meromorphic functions defined on this limited domain. However, the difference equations in $\nu$ can then be used to show that: (i) the limit in (4.2) exists for all $\nu$ in the whole complex plane away from a discrete subset, (ii) $\mathcal {H}$ is in fact a meromorphic function of $(\sigma _s, \nu )$ in all of $\mathbb {C}^2$, and (iii) the four difference equations hold as equalities between meromorphic functions on $\mathbb {C}^2$, see Proposition 4.5.

4.2.1 First Pair of Difference Equations

Define the difference operator $D_\mathcal {H}(b,\sigma _s)$ by

$$\begin{aligned} \begin{aligned} D_\mathcal {H}(b,\sigma _s)&= d^+_\mathcal {H}(b,\sigma _s) e^{ib\partial _{\sigma _s}} + d^+_\mathcal {H}(b,-\sigma _s) e^{-ib\partial _{\sigma _s}} + d^0_\mathcal {H}(b,\sigma _s), \end{aligned}\end{aligned}$$

(4.23)

where $d^0_\mathcal {H}$ is defined by

$$\begin{aligned} \begin{aligned} d^0_\mathcal {H}(b,\sigma _s)&=e^{- \pi b(iQ+\theta _*+2\theta _t)}-d^+_\mathcal {H}(b,\sigma _s)-d^+_\mathcal {H}(b,-\sigma _s), \end{aligned} \end{aligned}$$

(4.24)

with

$$\begin{aligned}{} & {} d^+_\mathcal {H}(b,\sigma _s) \nonumber \\{} & {} \quad = -2 e^{-\pi b\left( \sigma _s+\frac{i b}{2}\right) }\cosh (\pi b (\tfrac{i b}{2}+\theta _*+\sigma _s))\frac{\prod _{\epsilon =\pm 1} \cosh \left( \pi b \left( \frac{ib}{2}+\theta _t+\sigma _s+\epsilon \theta _0 \right) \right) }{\sinh (\pi b (2 \sigma _s+i b)) \sinh (2 \pi b \sigma _s)}\nonumber \\ \end{aligned}$$

(4.25)

Proposition 4.3

For $\sigma _s \in \mathbb {C}$ and $\text {Im} \,\nu > -Q/2$, the function $\mathcal {H}$ defined by (4.2) satisfies the following pair of difference equations:

$$\begin{aligned}&D_\mathcal {H}(b,\sigma _s) ~ \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = e^{-2\pi b \nu } \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ), \end{aligned}$$

(4.26a)

$$\begin{aligned}&D_\mathcal {H}(b^{-1},\sigma _s) ~ \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = e^{-2\pi b^{-1} \nu } \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ). \end{aligned}$$

(4.26b)

Proof

The proof consists of taking the confluent limit (4.2) of the difference equation (3.9a). On the one hand, we have

$$\begin{aligned} \lim \limits _{\Lambda \rightarrow -\infty } \Big (e^{\pi b \Lambda }2\cosh {2\pi b \sigma _t}|_{\sigma _t = \tfrac{\Lambda }{2}+\nu }\Big ) = e^{-2\pi b \nu }. \end{aligned}$$

(4.27)

On the other hand, straightforward computations using asymptotics of hyperbolic functions show that the following limits hold:

$$\begin{aligned}{} & {} \lim \limits _{\Lambda \rightarrow -\infty } e^{\pi b \Lambda } d_\mathcal {R}^+\left[ \begin{array}{c} \tfrac{\Lambda +\theta _*}{2}\;\;\theta _t\vspace{0.1cm}\\ \tfrac{\Lambda -\theta _*}{2} \;\theta _0 \end{array};\begin{array}{c} b,\;\pm \sigma _s \end{array}\right] = d^+_\mathcal {H}(b,\pm \sigma _s), \qquad \nonumber \\{} & {} \quad \lim \limits _{\Lambda \rightarrow -\infty } e^{\pi b \Lambda } d_\mathcal {R}^0\left[ \begin{array}{c} \tfrac{\Lambda +\theta _*}{2}\;\;\theta _t\vspace{0.1cm}\\ \tfrac{\Lambda -\theta _*}{2} \;\theta _0 \end{array};\begin{array}{c} b,\;\sigma _s \end{array}\right] = d^0_\mathcal {H}(b,\sigma _s), \end{aligned}$$

(4.28)

where $d_\mathcal {R}^+$ and $d_\mathcal {R}^0$ are defined in (3.7) and (3.8), respectively. Therefore we obtain

$$\begin{aligned} \lim \limits _{\Lambda \rightarrow -\infty } e^{\pi b \Lambda } D_\mathcal {R}\left[ \begin{array}{c} \tfrac{\Lambda +\theta _*}{2}\;\;\theta _t\vspace{0.1cm}\\ \tfrac{\Lambda -\theta _*}{2} \;\theta _0 \end{array};\begin{array}{c} b,\;\sigma _s \end{array}\right] = D_\mathcal {H}(b,\sigma _s), \end{aligned}$$

(4.29)

where $D_\mathcal {R}$ is given in (3.6). By Theorem 4.2, the limit in (4.2) exists whenever $(\sigma _s, \nu )\in \Omega _{\mathcal {H}}$. Thus, the difference equation (4.26a) follows after multiplying (3.9a) by $e^{\pi b \Lambda }$ and utilizing (4.27), (4.29), and the definition (4.2) of $\mathcal {H}$. Finally, (4.26b) follows from (4.26a) and the symmetry (4.22) of $\mathcal {H}$. $\square$

4.2.2 Second Pair of Difference Equations

Define the dual difference operator $\tilde{D}_\mathcal {H}(b,\nu )$ by

$$\begin{aligned} \tilde{D}_\mathcal {H}(b,\nu ) = \tilde{d}_\mathcal {H}^+(b,\nu ) e^{ib\partial _\nu } + \tilde{d}_\mathcal {H}^-(b,\nu ) e^{-ib\partial _\nu } + \tilde{d}_\mathcal {H}^0(b,\nu ), \end{aligned}$$

(4.30)

where $\tilde{H}_{\mathcal {H}}^0$ is defined by

$$\begin{aligned} \tilde{d}_{\mathcal {H}}^0(b,\nu ) = -2{\text {cosh}}{(2\pi b \left( \tfrac{ib}{2}+\theta _0+\theta _t \right) )}-\tilde{d}^+_\mathcal {H}(b,\nu )-\tilde{d}^{-}_\mathcal {H}(b,\nu ), \end{aligned}$$

(4.31)

with

$$\begin{aligned} \tilde{d}^\pm _\mathcal {H}(b,\nu )= & {} -4e^{2\pi b \nu }e^{\mp \pi b(\theta _0+\theta _t)} {\text {cosh}}{\left( \pi b \left( \tfrac{ib}{2}+\theta _0 \pm (\nu + \tfrac{\theta _*}{2}) \right) \right) }\nonumber \\{} & {} \times {\text {cosh}}{\left( \pi b \left( \tfrac{ib}{2}+\theta _t \pm (\nu - \tfrac{\theta _*}{2}) \right) \right) }. \end{aligned}$$

(4.32)

Proposition 4.4

For $\sigma _s \in \mathbb {C}$ and $\text {Im} \,(\nu - ib^{\pm 1}) > -Q/2$, the function $\mathcal {H}$ satisfies the following pair of difference equations:

$$\begin{aligned}&\tilde{D}_\mathcal {H}(b,\nu ) ~ \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = 2\cosh {(2\pi b \sigma _s)} ~ \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ), \end{aligned}$$

(4.33a)

$$\begin{aligned}&\tilde{D}_\mathcal {H}(b^{-1},\nu ) ~ \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = 2\cosh {(2\pi b^{-1} \sigma _s)} ~ \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ). \end{aligned}$$

(4.33b)

Proof

It is straightforward to verify that the following limits hold:

$$\begin{aligned}{} & {} \lim \limits _{\Lambda \rightarrow -\infty } d_\mathcal {R}^+\left[ \begin{array}{c} \theta _0\;\;\;\;\;\;\theta _t\vspace{0.1cm}\\ \frac{\Lambda -\theta _*}{2}\;\frac{\Lambda +\theta _*}{2} \end{array};\begin{array}{c} b,\;\pm (\frac{\Lambda }{2}+\nu ) \end{array}\right] = \tilde{d}^\pm _\mathcal {H}(b,\nu ), \nonumber \\{} & {} \quad \quad \lim \limits _{\Lambda \rightarrow -\infty } d_\mathcal {R}^0\left[ \begin{array}{c} \theta _0\;\;\;\;\;\;\theta _t\vspace{0.1cm}\\ \frac{\Lambda -\theta _*}{2}\;\frac{\Lambda +\theta _*}{2} \end{array};\begin{array}{c} b,\;\frac{\Lambda }{2}+\nu \end{array}\right] = \tilde{d}^0_\mathcal {H}(b,\nu ), \end{aligned}$$

(4.34)

where $d_\mathcal {R}^+$ and $d_\mathcal {R}^0$ are defined in (3.7) and (3.8), respectively. We obtain

$$\begin{aligned} \lim \limits _{\Lambda \rightarrow -\infty } D_\mathcal {R}\left[ \begin{array}{c} \theta _0\;\;\;\;\;\;\theta _t\vspace{0.1cm}\\ \frac{\Lambda -\theta _*}{2}\;\frac{\Lambda +\theta _*}{2} \end{array};\begin{array}{c} b,\;\frac{\Lambda }{2}+\nu \end{array}\right] = \tilde{D}_\mathcal {H}(b,\nu ), \end{aligned}$$

(4.35)

where $D_\mathcal {R}$ is defined in (3.6). The difference equation (4.33a) follows from (3.9c), (4.2), (4.35), and Theorem 4.2. Finally, (4.33b) follows from (4.33a) and the symmetry (4.22) of $\mathcal {H}$. $\square$

Using the difference equations (4.33), we can show that $\mathcal {H}$ extends to a meromorphic function of $\nu$ everywhere in the complex plane. More precisely, we have the following proposition.

Proposition 4.5

Let $(b,\theta _0,\theta _t,\theta _*) \in (0,\infty ) \times \mathbb {R}^3$ and $\sigma _s \in \mathbb {C} {\setminus } \Delta _\mathcal {H,\sigma _s}$. Then there is a discrete subset $\Delta \subset \mathbb {C}$ such that the limit in (4.2) exists for all $\nu \in \mathbb {C}\setminus \Delta$. Moreover, the function $\mathcal {H}$ defined by (4.2) has a meromorphic continuation to $(\sigma _s, \nu ) \in \mathbb {C}^2$ and the four difference equations (4.26) and (4.33) hold as equalities between meromorphic functions of $(\sigma _s, \nu ) \in \mathbb {C}^2$.

Proof

Consider $\nu$ such that $\text {Im} \,\nu > -Q/2$ but $\text {Im} \,(\nu -ib) \le -Q/2$. Solving (3.9c) for

$$\begin{aligned} e^{-ib\partial _\nu }\bigg (\mathcal {R}\left[ \begin{array}{c} \frac{\Lambda +\theta _*}{2}\;\; \theta _t\vspace{0.1cm}\\ \frac{\Lambda -\theta _*}{2}\;\; \theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \frac{\Lambda }{2}+\nu \end{array},b\right] \bigg ), \end{aligned}$$

taking the confluent limit $\Lambda \rightarrow -\infty$, and using (4.2), (4.35), and Theorem 4.2, we conclude that the limit in (4.2) exists also for $\nu$ in the strip $\{\nu \in \mathbb {C} | -b -Q/2 < \text {Im} \,\nu \le -Q/2\} {\setminus } \Delta _1$, where $\Delta _1$ is a discrete set. By iteration, we conclude that the limit in (4.2) exists for all $\nu \in \mathbb {C} {\setminus } \Delta$, where $\Delta$ is a discrete set. This proves the first assertion. The remaining assertions now follow by repeating the proofs of Propositions 4.3 and 4.4 with $\nu \in \mathbb {C}{\setminus } \Delta$. $\square$

4.3 First Polynomial Limit

In this subsection, we show that the function $\mathcal {H}$ reduces to the continuous dual q-Hahn polynomials when $\nu$ is suitably discretized. In addition to Assumption 1.1, we make the following assumption.

Assumption 4.6

(Restriction on the parameters) Assume that $b > 0$ is such that $b^2$ is irrational, and that

$$\begin{aligned} \begin{aligned}&\theta _t, \text {Re} \,\sigma _s \ne 0, \qquad \text {Re} \,\big (\tfrac{\theta _*}{2}- \nu -\theta _0 \pm \sigma _s \big ) \ne 0, \qquad \theta _0 + \theta _* \pm \theta _t \ne 0. \end{aligned}\end{aligned}$$

(4.36)

Assumption 4.6 implies that the three increasing and the three decreasing sequences of poles of the integrand in (4.5) do not overlap. The assumption that $b^2$ is irrational ensures that all the poles of the integrand are simple and that $q=e^{2i\pi b^2}$ is not a root of unity.

Define $\{\nu _n\}_{n=0}^\infty \subset \mathbb {C}$ by

$$\begin{aligned} \nu _n = \tfrac{\theta _*}{2}+\theta _t+\tfrac{i Q}{2}+inb. \end{aligned}$$

(4.37)

The sequence $\{\nu _n\}_{n=0}^\infty$ is a subset of the set $\Delta _\nu$ of possible poles of $\mathcal {H}$ defined in (4.4b). The following theorem shows that $\mathcal {H}$ still has a finite limit as $\nu \rightarrow \nu _n$ for each $n \ge 0$ and that the limit is given by the continuous dual q-Hahn polynomials. The reason the limit is finite is that the prefactor $P_\mathcal {H}$ has a simple zero at each $\nu _n$; this zero cancels the simple pole that the integral in (4.5) has due to the contour of integration being pinched between two poles of the integrand.

Theorem 4.7

(From $\mathcal {H}$ to the continuous dual q-Hahn polynomials) Let $\sigma _s \in \mathbb {C} \setminus \Delta _\mathcal {H,\sigma _s}$ and suppose that Assumptions 1.1 and 4.6 are satisfied. Under the parameter correspondence

$$\begin{aligned} \alpha _H=e^{2\pi b(\theta _t+\theta _0+\frac{iQ}{2})}, \quad \beta _{H}=e^{2\pi b (\theta _t-\theta _0+\frac{iQ}{2})}, \quad \gamma _{H}=e^{2\pi b (\theta _*+\frac{iQ}{2})}, \qquad q=e^{2i\pi b^2},\nonumber \\ \end{aligned}$$

(4.38)

the function $\mathcal {H}$ defined in (4.5) satisfies, for each integer $n\ge 0$,

$$\begin{aligned} \lim \limits _{\nu \rightarrow \nu _n} \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = H_n(e^{2\pi b \sigma _s};\alpha _H,\beta _{H},\gamma _{H},q), \end{aligned}$$

(4.39)

where $H_n$ are the continuous dual q-Hahn polynomials defined in (B.8).

Proof

There are two different ways to prove (4.39). The first approach consists of taking the limit $\nu \rightarrow \nu _n$ in the integral representation (4.5) for $\mathcal {H}$ for each n; the second approach computes the limit for $n=0$ and then uses the limit of the difference equation (4.33a) to extend the result to other values of n. The first approach is described in detail in Sect. 10 for the function $\mathcal {M}$. Here we use the second approach.

We first show that the limit in (4.39) exists for $n=0$ and equals 1. The function $s_b(\nu -\tfrac{\theta _*}{2}-\theta _t)$ in (4.6) has a simple zero at $\nu _0=\tfrac{\theta _*}{2}+\theta _t+\tfrac{iQ}{2}$. Moreover, in the limit $\nu \rightarrow \nu _0$, the pole of the function $s_b(x+\tfrac{\theta _*}{2}+\theta _t-\nu )$ in (4.7) at $x_0:=-\tfrac{iQ}{2}-\tfrac{\theta _*}{2}-\theta _t+\nu$ collides with the pole of $s_b(x+\tfrac{iQ}{2})^{-1}$ located at $x=0$, pinching the contour $\mathcal {C}_{\mathcal {H}}$. Therefore, before taking the limit $\nu \rightarrow \nu _0$, we deform the contour $\mathcal {C}_{\mathcal {H}}$ into a contour $\mathcal {C}_{\mathcal {H}}'$ which passes below $x_0$. We obtain

$$\begin{aligned}{} & {} \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = -2i\pi \; P_\mathcal {H}\left( \sigma _s,\nu \right) \underset{x=x_0}{\text {Res}}\left( I_\mathcal {H}(x,\sigma _s,\nu )\right) \nonumber \\{} & {} \quad + P_\mathcal {H}(\sigma _s,\nu ) \displaystyle \int _{\mathcal {C}_{\mathcal {H}}'} dx \; I_\mathcal {H}(x,\sigma _s,\nu ). \end{aligned}$$

(4.40)

Using the residue (2.4), a straightforward computation yields

$$\begin{aligned} -2i\pi \underset{x=x_0}{\text {Res}}\left( I_\mathcal {H}(x,\sigma _s,\nu )\right) =&\; e^{i \pi \left( \nu -\theta _0+\frac{\theta _*}{2} -\frac{i Q}{2}\right) \left( \nu -\frac{\theta _*}{2}-\theta _t -\frac{i Q}{2}\right) } \nonumber \\&\times \frac{s_b\left( \theta _0-\frac{\theta _*}{2}+\nu -\frac{i Q}{2}-\sigma _s\right) s_b\left( \theta _0-\frac{\theta _*}{2}+\nu -\frac{i Q}{2}+\sigma _s\right) }{s_b\left( \theta _0+\frac{\theta _*}{2}+\nu \right) s_b\left( \nu -\frac{\theta _*}{2}-\theta _t\right) s_b\left( \nu -\frac{\theta _*}{2}+\theta _t \right) }. \end{aligned}$$

(4.41)

The right-hand side of (4.41) has a simple pole at $\nu =\nu _0$ due to the factor $s_b(\nu -\tfrac{\theta _*}{2}-\theta _t)^{-1}$. Moreover, in the limit $\nu \rightarrow \nu _0$ the second term in (4.40) vanishes thanks to the zero of $P_\mathcal {H}$. Thus we obtain

$$\begin{aligned} \lim \limits _{\nu \rightarrow \nu _0} \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = -2i\pi \lim \limits _{\nu \rightarrow \nu _0}\Big ( P_\mathcal {H}\left( \sigma _s,\nu \right) \underset{x=x_0}{\text {Res}}\left( I_\mathcal {H}(x,\sigma _s,\nu )\right) \Big ). \end{aligned}$$

(4.42)

Using $s_b(x)=s_b(-x)^{-1}$, it is straightforward to verify that the right-hand side equals 1; this proves (4.39).

For each integer $n\ge 0$, let $P_n$ denote the left-hand side of (4.39):

$$\begin{aligned} P_n:= \lim \limits _{\nu \rightarrow \nu _n} \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ). \end{aligned}$$

(4.43)

The same kind of contour deformation used to establish the case $n=0$ shows that the limit in (4.43) exists for all $n \ge 0$. To show that $P_n$ equals the continuous dual q-Hahn polynomials $H_n$, we consider the limit $\nu \rightarrow \nu _n$ of the difference equation (4.33a). Using the parameter correspondence (4.38), it is straightforward to verify that

$$\begin{aligned} \lim \limits _{\nu \rightarrow \nu _n} \tilde{D}_\mathcal {H}(b,\nu ) = L_H, \end{aligned}$$

(4.44)

where the operators $\tilde{D}_\mathcal {H}$ and $L_H$ are defined in (4.30) and (B.10), respectively. Hence taking the limit $\nu \rightarrow \nu _n$ of the difference equation (4.33a), we see that $P_n$ satisfies

$$\begin{aligned} L_H P_n = \left( z+z^{-1} \right) P_n, \end{aligned}$$

(4.45)

where $z=e^{2\pi b \sigma _s}$. Thus the $P_n$ satisfy the same recurrence relation (B.9) as the continuous dual q-Hahn polynomials evaluated at $z=e^{2\pi b \sigma _s}$. Since we have already shown that $P_0=H_0=1$, we infer that $P_n=H_n$ for all $n\ge 0$ by induction, where $H_n$ is evaluated at $z=e^{2\pi b \sigma _s}$. This completes the proof of (4.39). $\square$

4.4 Second Polynomial Limit

In this subsection, we show that $\mathcal {H}$ reduces to the big q-Jacobi polynomials when $\sigma _s$ is suitably discretized.

Theorem 4.8

(From $\mathcal {H}$ to the big q-Jacobi polynomials) Let $\nu \in \{\text {Im} \,\nu > -Q/2\} \setminus \Delta _{\nu }$ and suppose that Assumptions 1.1 and 4.6 are satisfied. Under the parameter correspondence

$$\begin{aligned}{} & {} \alpha _J = e^{4\pi b \theta _t}, \quad \beta _J = e^{4\pi b \theta _0}, \quad \gamma _J = e^{2\pi b(\theta _0+\theta _*+\theta _t)}, \quad x_J=e^{2\pi b (\theta _t+\frac{\theta _*}{2}+\frac{iQ}{2})}e^{-2\pi b \nu },\nonumber \\{} & {} \quad q=e^{2i\pi b^2}, \end{aligned}$$

(4.46)

the function $\mathcal {H}$ defined in (4.5) satisfies, for each integer $n\ge 0$,

$$\begin{aligned} \lim \limits _{\sigma _s \rightarrow \sigma _s^{(n)}} \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = J_n(x_J;\alpha _J,\beta _J,\gamma _J;q), \end{aligned}$$

(4.47)

where $\sigma _s^{(n)} \in \mathbb {C}$ is given in (3.11) and where $J_n$ are the big q-Jacobi polynomials defined in (B.12).

Proof

We first prove that (4.47) holds for $n=0$. The function $s_b(\sigma _s-\theta _0-\theta _t)$ in (4.6) has a simple zero located at $\sigma _s=\sigma _s^{(0)}=\theta _0+\theta _t+\tfrac{iQ}{2}$. On the other hand, in the limit $\sigma _s=\sigma _s^{(0)}$, the contour $\mathcal {C}_{\mathcal {H}}$ is squeezed between the pole of $s_b(x+\theta _0+\theta _t-\sigma _s)$ in (4.7) located at $x_0=-\tfrac{iQ}{2}+\sigma _s-\theta _0-\theta _t$ and the pole of $s_b(x+\tfrac{iQ}{2})^{-1}$ at $x=0$. Therefore, before taking the limit $\sigma _s \rightarrow \sigma _s^{(0)}$, we deform the contour $\mathcal {C}_{\mathcal {H}}$ into a contour $\mathcal {C}_{\mathcal {H}}'$ which passes below $x_0$. We obtain

$$\begin{aligned}{} & {} \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = -2i\pi \; P_\mathcal {H}\left( \sigma _s,\nu \right) \underset{x=x_0}{\text {Res}}\left( I_\mathcal {H}(x,\sigma _s,\nu )\right) \nonumber \\{} & {} \quad + P_\mathcal {H}\left( \sigma _s,\nu \right) \displaystyle \int _{\mathcal {C}_{\mathcal {H}}'} dx \; I_\mathcal {H}(x,\sigma _s,\nu ). \end{aligned}$$

(4.48)

A straightforward computation using (2.4) shows that

$$\begin{aligned}&-2i\pi \underset{x=x_0}{\text {Res}} I_\mathcal {H}(x,\sigma _s,\nu ) \\&\quad = e^{i \pi \left( -\theta _0+\frac{\theta _*}{2}+\nu -\frac{i Q}{2}\right) \left( -\theta _0-\theta _t-\frac{i Q}{2}+\sigma _s\right) } \frac{s_b\left( 2 \sigma _s-\frac{i Q}{2}\right) s_b\left( \sigma _s-\theta _0+\frac{\theta _*}{2}-\nu -\frac{i Q}{2}\right) }{s_b(\theta _*+\sigma _s) s_b(\sigma _s-\theta _0-\theta _t) s_b(\sigma _s-\theta _0+\theta _t)}. \end{aligned}$$

The right-hand side has a simple pole at $\sigma _s=\sigma _s^{(0)}$ due to the factor $s_b(\sigma _s-\theta _0-\theta _t)^{-1}$. Moreover, the second term in (4.48) vanishes at $\sigma _s=\sigma _s^{(0)}$ thanks to the zero of $P_\mathcal {H}$. Therefore,

$$\begin{aligned} \lim \limits _{\sigma _s\rightarrow \sigma _s^{(0)}} \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = -2i\pi \lim \limits _{\sigma _s\rightarrow \sigma _s^{(0)}} \Big (P_\mathcal {H}\left( \sigma _s,\nu \right) \underset{x=x_0}{\text {Res}}\left( I_\mathcal {H}(x,\sigma _s,\nu )\right) \Big ). \nonumber \\\end{aligned}$$

(4.49)

A straightforward computation using $s_b(x)=s_b(-x)^{-1}$ shows that the right-hand side equals 1; this proves (4.47) for $n=0$.

We now use the difference equation (4.26a) to show that (4.47) holds also for $n \ge 1$. Let

$$\begin{aligned} P_n:= \lim \limits _{\sigma _s\rightarrow \sigma _s^{(n)}} \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ), \qquad n = 0, 1, \dots . \end{aligned}$$

(4.50)

The same kind of contour deformation used for $n=0$ shows that the limit in (4.47) exists for all $n \ge 0$. Moreover, under the parameter correspondence (4.46), we have

$$\begin{aligned} \lim \limits _{\sigma _s\rightarrow \sigma _s^{(n)}} D_\mathcal {H}(b,\sigma _s) = e^{-2\pi b(\theta _t+\frac{\theta _*}{2}+\frac{iQ}{2})} L_J, \end{aligned}$$

(4.51)

where the operators $D_\mathcal {H}$ and $L_J$ are defined in (4.23) and (B.14), respectively. Hence taking the limit $\sigma _s\rightarrow \sigma _s^{(n)}$ of (4.26a) and recalling that $x_J =e^{2\pi b (\theta _t+\frac{\theta _*}{2}+\frac{iQ}{2})}e^{-2\pi b \nu }$, we see that $P_n$ satisfies $L_J P_n = x_J P_n$. Thus the $P_n$ satisfy the same recurrence relation (B.13) as the big q-Jacobi polynomials $J_n$, and the limit (4.47) for $n\ge 1$ follows by induction. $\square$

Remark 4.9

It is also possible to define a function $\mathcal {H}'$ by sending $\Lambda \rightarrow +\infty$ in (4.2) instead of $\Lambda \rightarrow -\infty$:

$$\begin{aligned} \mathcal {H}'(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = \lim \limits _{\Lambda \rightarrow +\infty } \mathcal {R}\left[ \begin{array}{c} \frac{\Lambda +\theta _*}{2}\;\; \theta _t\vspace{0.1cm}\\ \frac{\Lambda -\theta _*}{2}\;\; \theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \frac{\Lambda }{2}+\nu \end{array}\right] . \end{aligned}$$

(4.52)

It can be shown that the limit in (4.52) exists for $(\sigma _s,\nu ) \in \Omega _{\mathcal {H}}$. Moreover, due to the asymptotic formula (2.8) for $s_b$, the only difference between $\mathcal {H}'$ and $\mathcal {H}$ resides in the sign of the phases in the representation (4.5). In fact, the following two limits hold:

$$\begin{aligned}&\lim \limits _{\nu \rightarrow \nu _n} \mathcal {H}'(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = H_n(e^{2\pi b \sigma _s};\alpha _H^{-1},\beta _{H}^{-1},\gamma _{H}^{-1},q^{-1}), \end{aligned}$$

(4.53)

$$\begin{aligned}&\lim \limits _{\sigma _s \rightarrow \sigma _s^{(n)}} \mathcal {H}'(b,\theta _0,\theta _t,\theta _*,\sigma _s,\nu ) = J_n(x_J^{-1};\alpha _J^{-1},\beta _J^{-1},\gamma _J^{-1};q^{-1}). \end{aligned}$$

(4.54)

We expect that a similar phenomenon is present for all families of the non-polynomial scheme. For simplicity, we will only study one of the two representatives for each family.

5 The Function $\mathcal {S}$

In this section, we introduce the function $\mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho )$ which is one of the two elements at the third level of the non-polynomial scheme, see Fig. 2. The function $\mathcal {S}$ is defined as a confluent limit of the function $\mathcal {H}$. We show that $\mathcal {S}$ is a joint eigenfunction of four difference operators and that it reduces to the Al-Salam Chihara and the little q-Jacobi polynomials, which lie at the third level of the q-Askey scheme, when $\rho$ and $\sigma _s$ are suitably discretized, respectively.

5.1 Definition and Integral Representation

Let $\rho$ be a new parameter defined in terms of $\theta _*$ and $\nu$ by

$$\begin{aligned} \nu = \frac{\theta _*}{2}+\rho . \end{aligned}$$

(5.1)

Define the open set $\Omega _\mathcal {S} \subset \mathbb {C}^2$ by

$$\begin{aligned} \Omega _\mathcal {S}:= (\mathbb {C} \backslash \Delta _{\mathcal S,\sigma _s}) \times (\{\text {Im} \,\rho > -Q/2\} \backslash \Delta _\rho ), \end{aligned}$$

(5.2)

where the discrete subsets $\Delta _\mathcal {S,\sigma _s}$ and $\Delta _\rho$ are given by

$$\begin{aligned}&\Delta _\mathcal {S,\sigma _s} := \{\pm \sigma _s \, |\, \sigma _s \in \Delta _{\mathcal {S}, \sigma _s}'\}, \\&\Delta _\rho := \{\pm \theta _t+\tfrac{iQ}{2}+ibm+ilb^{-1} \}_{m,l=0}^\infty \cup \{\theta _t-\tfrac{iQ}{2}-ibm-ilb^{-1} \}_{m,l=0}^\infty , \end{aligned}$$

with

$$\begin{aligned} \Delta _\mathcal {S,\sigma _s}' :=&\; \{\theta _0 \pm \theta _t + \tfrac{iQ}{2} + i m b +il b^{-1}\}_{m,l=0}^\infty \cup \{\theta _0+ \theta _t -\tfrac{i Q}{2} -i m b-il b^{-1}\}_{m,l=0}^\infty . \end{aligned}$$

Definition 5.1

Let $\mathcal {H}$ be defined by (4.2). The function $\mathcal {S}$ is defined for $(\sigma _s,\rho ) \in \Omega _{\mathcal {S}}$ by

$$\begin{aligned} \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ) = \lim \limits _{\theta _* \rightarrow -\infty } \mathcal {H}\left( b,\theta _0,\theta _t,\theta _*,\sigma _s,\tfrac{\theta _*}{2}+\rho \right) . \end{aligned}$$

(5.3)

The next theorem shows that $\mathcal {S}$ is a well-defined meromorphic function of $(\sigma _s,\rho ) \in \Omega _{\mathcal {S}}$, which has a meromorphic continuation to all of $\mathbb {C}^2$.

Theorem 5.2

Suppose that Assumption 1.1 is satisfied. The limit in (5.3) exists uniformly for $(\sigma _s,\rho )$ in compact subsets of $\Omega _{\mathcal {S}}$. Moreover, the function $\mathcal {S}$ can be extended to an analytic function of $(\sigma _s,\rho ) \in (\mathbb {C} \backslash \Delta _{\mathcal S,\sigma _s}) \times (\mathbb {C}\backslash \Delta _\rho )$ and admits the following integral representation:

$$\begin{aligned}{} & {} \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ) \nonumber \\{} & {} \quad = P_\mathcal {S}(\sigma _s,\rho ) \int _{\mathcal {C}_{\mathcal {S}}} dx~I_\mathcal {S}(x,\sigma _s,\rho ) \qquad \text {for }(\sigma _s,\rho ) \in (\mathbb {C} \backslash \Delta _\mathcal {S,\sigma _s}) \times (\mathbb {C}\backslash \Delta _\rho ),\nonumber \\ \end{aligned}$$

(5.4)

where the dependence of $P_\mathcal {S}$ and $I_\mathcal {S}$ on $b,\theta _0,\theta _t$ is omitted for simplicity,

$$\begin{aligned}&P_\mathcal {S}(\sigma _s,\rho ) = s_b\left( 2 \theta _t+\tfrac{i Q}{2}\right) s_b\left( \rho - \theta _t \right) \prod _{\epsilon =\pm 1} s_b \left( \epsilon \sigma _s -\theta _0-\theta _t\right) , \end{aligned}$$

(5.5)

$$\begin{aligned}&I_\mathcal {S}(x,\sigma _s,\rho ) = e^{-\frac{i\pi x^2}{2} -i \pi x \left( iQ+2\theta _0+\theta _t-\rho \right) } \frac{s_b(x+\theta _t-\rho ) }{s_b\left( x+\frac{i Q}{2}\right) s_b\left( x+\frac{i Q}{2}+2 \theta _t\right) }\nonumber \\&\quad ~~~~~~~~~~~~~~~~~~~~~~\times \prod _{\epsilon =\pm 1} s_b(x+\theta _0+\theta _t+\epsilon \sigma _s), \end{aligned}$$

(5.6)

and the contour $\mathcal {C}_{\mathcal {S}}$ is any curve from $-\infty$ to $+\infty$ which separates the three decreasing from the two increasing sequences of poles, with the requirement that its right tail satisfies

$$\begin{aligned} \text {Im} \,x -\text {Im} \,\rho < -\delta \qquad \text {for all }x \in \mathcal {C}_{\mathcal {S}}\text { with } \text {Re} \,x\text { sufficiently large}, \end{aligned}$$

(5.7)

for some $\delta > 0$. In particular, $\mathcal {S}$ is a meromorphic function of $(\sigma _s,\rho ) \in \mathbb {C}^2$. If $(\sigma _s,\rho ) \in \mathbb {R}^2$, then the contour $\mathcal C_\mathcal {S}$ can be any curve from $-\infty$ to $+\infty$ lying within the strip $\text {Im} \,x \in (-Q/2,-\delta )$.

Proof

The proof is similar to the proof of Theorem 4.2, but there are some differences because the exponent in (5.6) is a quadratic (rather than a linear) polynomial in x. Let $(b,\theta _0,\theta _t) \in (0,\infty ) \times \mathbb {R}^2$. It can be verified that

$$\begin{aligned} P_\mathcal {H}\left( \sigma _s,\tfrac{\theta _*}{2}+\rho \right) I_\mathcal {H}\left( x,\sigma _s,\tfrac{\theta _*}{2}+\rho \right) = P_\mathcal {S}(\sigma _s,\rho ) Z(x,\theta _*) I_\mathcal {S}(x,\sigma _s,\rho ), \end{aligned}$$

(5.8)

where

$$\begin{aligned} Z(x,\theta _*) = e^{\frac{i \pi x^2}{2}} e^{i \pi x \left( \theta _0+\theta _*+\theta _t+\frac{i Q}{2}\right) } \frac{s_b\left( \theta _0+\theta _*+\theta _t+\frac{i Q}{2}\right) }{s_b\left( x+\theta _0+\theta _*+\theta _t+\frac{i Q}{2}\right) }. \end{aligned}$$

(5.9)

Due to the properties (2.3) of the function $s_b$, the function $I_\mathcal {S}(\cdot ,\sigma _s,\rho )$ possesses two increasing sequences of poles starting at $x=0$ and $x=-2\theta _t$, as well as three decreasing sequences of poles starting at $x=-\tfrac{iQ}{2}+\rho -\theta _t$ and $x=-\tfrac{iQ}{2}\pm \sigma _s-\theta _0-\theta _t$. The discrete sets $\Delta _{\mathcal S,\sigma _s}$ and $\Delta _\rho$ contain all the values of $\sigma _s$ and $\rho$, respectively, for which poles in any of the two increasing sequences collide with poles in any of the three decreasing sequences. The sets $\Delta _\mathcal {S,\sigma _s}$ and $\Delta _\rho$ also contain all the values of $\sigma _s$ and $\rho$ at which the prefactor $P_\mathcal {S}(\sigma _s,\rho )$ has poles. Furthermore, $Z(x,\theta _*)$ possesses one increasing sequence of poles starting at $x=-\theta _0-\theta _*-\theta _t$ which lies in the half-plane $\text {Im} \,x \ge 0$. The real parts of the poles in this sequence tend to $+\infty$ as $\theta _*\rightarrow -\infty$.

Let $K_{\sigma _s}$ and $K_\rho$ be compact subsets of $\mathbb C \backslash \Delta _\mathcal {S,\sigma _s}$ and $\{\text {Im} \,\rho > -Q/2\} \backslash \Delta _\rho$, respectively. Suppose $(\sigma _s,\rho ) \in K_{\sigma _s} \times K_\rho$. Then, the above discussion shows that it is possible to choose a contour $\mathcal {C}_{\mathcal {S}} = \mathcal {C}_{\mathcal {S}}(\sigma _s,\rho )$ from $-\infty$ to $+\infty$ which separates the two upward from the three downward sequences of poles of $Z(\cdot ,\theta _*) I_\mathcal {S}(\cdot ,\sigma _s,\rho )$. Let us choose $\mathcal {C}_{\mathcal {S}}$ so that its right tail approaches the horizontal line $\text {Im} \,x = -Q/2-\delta$ as $\text {Re} \,x \rightarrow +\infty$ for some $\delta >0$. Then there is an $N>0$ such that $\mathcal {C}_{\mathcal {S}}$ is independent of $\theta _*$ for $\theta _*<-N$, and (4.5) and (5.8) imply that, for all $(\sigma _s,\rho ) \in K_{\sigma _s} \times K_\rho$ and all $\theta _* < -N$,

$$\begin{aligned} \mathcal {H}\left( b,\theta _0,\theta _t,\theta _*,\sigma _s,\tfrac{\theta _*}{2}+\rho \right) = P_\mathcal {S}(\sigma _s,\rho ) \displaystyle \int _{\mathcal {C}_{\mathcal {S}}} dx \; Z(x,\theta _*) I_\mathcal {S}(x,\sigma _s,\rho ). \end{aligned}$$

(5.10)

Utilizing the asymptotic formula (2.8), it can be verified that the following limit

$$\begin{aligned} \lim \limits _{\theta _* \rightarrow -\infty } Z(x,\theta _*) = 1, \end{aligned}$$

(5.11)

holds uniformly for $(\sigma _s,\rho ) \in K_{\sigma _s} \times K_\rho$ and for x in bounded subsets of $\mathcal {C}_{\mathcal {S}}$. Moreover, using (2.8) with $\epsilon =1/2$ we find that $I_\mathcal {S}$ obeys the estimates

$$\begin{aligned} I_\mathcal {S}(x,\sigma _s,\rho ) = {\left\{ \begin{array}{ll} O\left( e^{2\pi |\text {Re} \,x |(\text {Im} \,x-\text {Im} \,\rho )}\right) , &{} \text {Re} \,x \rightarrow +\infty , \\ O\left( e^{-2\pi Q |\text {Re} \,x|}\right) , &{} \text {Re} \,x \rightarrow -\infty , \end{array}\right. } \end{aligned}$$

(5.12)

uniformly for $(\sigma _s,\rho )$ in compact subsets of $K_{\sigma _s} \times K_\rho$ and $\text {Im} \,x$ in compact subsets of $\mathbb {R}$. Since the contour $\mathcal {C}_{\mathcal {S}}$ stays a bounded distance away from the poles of the integrand $I_\mathcal {S}$, we infer that there exists a constant $C_1>0$ such that

$$\begin{aligned} |I_\mathcal {S}(x,\sigma _s,\rho ) |\le C_1 \times {\left\{ \begin{array}{ll} \; e^{2\pi |\text {Re} \,x|(\text {Im} \,x-\text {Im} \,\rho )}, &{} \text {Re} \,x \ge 0, \\ \; e^{-2\pi Q |\text {Re} \,x|}, &{} \text {Re} \,x \le 0, \end{array}\right. } \end{aligned}$$

(5.13)

uniformly for $(\sigma _s,\rho )$ in compact subsets of $K_{\sigma _s} \times K_\rho$. In particular, the integrand $I_\mathcal {S}$ has exponential decay along the left and right tails of the contour $\mathcal {C}_{\mathcal {S}}$.

Suppose we can show that there exist constants $c>0$ and $C>0$ such that

$$\begin{aligned} |Z(x,\theta _*) I_\mathcal {S}(x,\sigma _s,\rho ) |\le C e^{-c |\text {Re} \,x |}, \end{aligned}$$

(5.14)

uniformly for all $\theta _*<-N$, $x\in \mathcal {C}_{\mathcal {S}}$ and $(\sigma _s,\rho )\in K_{\sigma _s} \times K_\rho$. Then it follows from (5.10), (5.11) and Lebesgue’s dominated convergence theorem that the limit in (5.3) exists uniformly for $(\sigma _s,\rho )\in K_{\sigma _s} \times K_\rho$ and is given by (5.4). This proves that the limit in (5.3) exists uniformly for $(\sigma _s,\rho )$ in compact subsets of $\Omega _{\mathcal {S}}$ and proves (5.4) for $(\sigma _s,\rho ) \in \Omega _{\mathcal {S}}$. The analyticity of $\mathcal S$ as a function of $(\sigma _s,\rho ) \in \Omega _{\mathcal {S}}$ follows from the analyticity of $\mathcal H$ together with the uniform convergence on compact subsets.

Let $c_0 < -Q/2$ be a constant. Then we can choose the contour $\mathcal {C}_{\mathcal {S}}$ so that it satisfies the condition in (5.7) for any $\rho \in \mathbb {C} \backslash \Delta _\rho$ with $\text {Im} \,\rho > c_0$. For such a choice of $\mathcal {C}_{\mathcal {S}}$, the estimate (5.13) ensures that the integral representation (5.4) provides a meromorphic continuation of $\mathcal {S}$ to $(\sigma _s,\rho ) \in \mathbb {C} \times \{\text {Im} \,\rho > c_0\}$ which is analytic for $(\sigma _s,\rho ) \in (\mathbb {C} \backslash \Delta _{\mathcal S,\sigma _s}) \times (\{\text {Im} \,\rho > c_0\} \backslash \Delta _\rho )$. Since $c_0$ was arbitrary, it follows that $\mathcal {S}$ admits a meromorphic continuation to $(\sigma _s,\rho ) \in \mathbb {C}^2$ which is analytic for $(\sigma _s,\rho ) \in (\mathbb {C} \backslash \Delta _\mathcal {S,\sigma _s}) \times (\mathbb {C}\backslash \Delta _\rho )$.

To complete the proof of the theorem, it only remains to prove (5.14). The asymptotic formula (2.8) for $s_b$ with $\epsilon =1/2$ implies that there exist constants $C_2,C_3,C_4>0$ such that the inequalitites

$$\begin{aligned}&|s_b\left( \theta _0+\theta _*+\theta _t+\tfrac{i Q}{2}\right) |\le C_2 \; e^{\frac{\pi Q |\theta _* |}{2}}, \qquad \theta _*<-N, \end{aligned}$$

(5.15)

$$\begin{aligned}&|s_b\left( x+\theta _0+\theta _*+\theta _t+\tfrac{i Q}{2}\right) ^{-1} |\le C_3 \; e^{-\frac{\pi Q |\theta _*+\text {Re} \,x |}{2}} e^{-\pi (\text {Im} \,x) |\theta _*+\text {Re} \,x |}, \qquad x \in \mathcal {C}_{\mathcal {S}}, \; \theta _* \in \mathbb {R}, \end{aligned}$$

(5.16)

$$\begin{aligned}&|e^{\frac{i \pi x^2}{2}} e^{i \pi x \left( \theta _0+\theta _*+\theta _t+\frac{i Q}{2}\right) } |\le C_4 \; e^{-\pi (\text {Im} \,x)(\theta _*+\text {Re} \,x)} e^{-\frac{\pi Q \text {Re} \,x}{2}}, \qquad x \in \mathcal {C}_{\mathcal {S}}, \end{aligned}$$

(5.17)

hold uniformly for $(\sigma _s,\rho )\in K_{\sigma _s} \times K_\rho$. Therefore, in view of (5.9), there exists a constant $C_5>0$ such that

$$\begin{aligned} |Z(x,\theta _*) |\le C_5 \; e^{-\pi (\text {Im} \,x+\frac{Q}{2})(\theta _*+\text {Re} \,x+|\theta _*+\text {Re} \,x |)}, \quad x \in \mathcal {C}_{\mathcal {S}}, \; \theta _*<-N, \end{aligned}$$

(5.18)

uniformly for $(\sigma _s,\rho )\in K_{\sigma _s} \times K_\rho$. The inequality (5.18) can be rewritten as follows:

$$\begin{aligned} |Z(x,\theta _*) |\le {\left\{ \begin{array}{ll} C_5 \; e^{-2\pi (\text {Im} \,x+\frac{Q}{2})(\theta _*+\text {Re} \,x)}, &{} \theta _* + \text {Re} \,x \ge 0, \\ C_5, &{} \theta _* + \text {Re} \,x \le 0, \end{array}\right. } \quad x \in \mathcal {C}_{\mathcal {S}}, ~ \theta _* < -N,\nonumber \\ \end{aligned}$$

(5.19)

uniformly for $(\sigma _s,\rho )\in K_{\sigma _s} \times K_\rho$. The inequalities (5.13) and (5.19) yield the existence of a constant $C_6>0$ independent of $x \in \mathcal {C}_{\mathcal {S}}$, $\theta _* < -N$ and $(\sigma _s,\rho )\in K_{\sigma _s} \times K_\rho$ such that

$$\begin{aligned} |Z(x,\theta _*) I_\mathcal {S}(x,\sigma _s,\rho ) |\le {\left\{ \begin{array}{ll} C_6 \; e^{-2\pi (\text {Re} \,x) (\frac{Q}{2}+\text {Im} \,\rho )} e^{-2\pi \theta _*(\frac{Q}{2}+\text {Im} \,x)}, &{} \text {Re} \,x \ge -\theta _*, \\ C_6 \; e^{2\pi (\text {Re} \,x)(\text {Im} \,x-\text {Im} \,\rho )}, &{} 0 \le \text {Re} \,x \le -\theta _*, \\ C_6 \; e^{-2\pi Q |\text {Re} \,x|}, &{} \text {Re} \,x \le 0, \end{array}\right. }\nonumber \\ \end{aligned}$$

(5.20)

uniformly for $(\sigma _s,\rho )\in K_{\sigma _s} \times K_\rho$. Since $\text {Im} \,\rho > -Q/2 + \delta$ for $\rho \in K_\rho$ and $\text {Im} \,x < -Q/2 -\delta$ on the right tail of the contour for some $\delta > 0$, this proves (5.14) and completes the proof. $\square$

Furthermore, thanks to the property (2.7) of $s_b$ the function $\mathcal {S}$ has the symmetry

$$\begin{aligned} \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho )=\mathcal {S}(b^{-1},\theta _0,\theta _t,\sigma _s,\rho ). \end{aligned}$$

(5.21)

5.2 Difference Equations

By taking the confluent limit (5.3) of the difference equations (4.26) and (4.33) satisfied by $\mathcal {H}$, it follows that the function $\mathcal {S}$ is a joint eigenfunction of four different difference operators, two acting on $\sigma _s$ and the remaining two acting on $\rho$. The four difference equations will hold as equalities between meromorphic functions on $\mathbb C^2$. Since the derivations are similar to those presented in Sect. 4.2, we state these results without proofs.

5.2.1 First Pair of Difference Equations

Define the difference operator $D_{\mathcal {S}}(b,\sigma _s)$ by

$$\begin{aligned} D_{\mathcal {S}}(b,\sigma _s) = d^+_{\mathcal {S}}(b,\sigma _s)e^{ib\partial _{\sigma _s}}+d^+_{\mathcal {S}}(b,-\sigma _s)e^{-ib\partial _{\sigma _s}}+d^0_{\mathcal {S}}(b,\sigma _s), \end{aligned}$$

(5.22)

where

$$\begin{aligned}&d^+_{\mathcal {S}}(b,\sigma _s) = e^{-\pi b (2 \sigma _s+i Q)} \frac{\prod _{\epsilon =\pm 1}\cosh \left( \pi b \left( \frac{ib}{2}+\epsilon \theta _0+\theta _t+\sigma _s\right) \right) }{\text {sinh}(\pi b (2 \sigma _s+i b)) \text {sinh}(2 \pi b \sigma _s)}, \end{aligned}$$

(5.23)

$$\begin{aligned}&d^0_{\mathcal {S}}(b,\sigma _s)=e^{-\pi b (2 \theta _t+i Q)}-d^+_{\mathcal {S}}(b,\sigma _s)-d^+_{\mathcal {S}}(b,-\sigma _s). \end{aligned}$$

(5.24)

Proposition 5.3

For $(\sigma _s,\rho )\in \mathbb C^2$, the function $\mathcal {S}$ satisfies the pair of difference equations

$$\begin{aligned}&D_{\mathcal {S}}(b,\sigma _s) \; \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ) = e^{-2\pi b \rho } \; \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ), \end{aligned}$$

(5.25a)

$$\begin{aligned}&D_{\mathcal {S}}(b^{-1},\sigma _s) \; \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ) = e^{-2\pi b^{-1} \rho } \; \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ). \end{aligned}$$

(5.25b)

5.2.2 Second Pair of Difference Equations

Define the difference operator $\tilde{D}_{\mathcal {S}}(b,\rho )$ such that

$$\begin{aligned} \tilde{D}_{\mathcal {S}}(b,\rho ) = \tilde{d}^+_{\mathcal {S}}(b,\rho ) e^{ib \partial _\rho } + \tilde{d}^-_{\mathcal {S}}(b,\rho ) e^{-ib \partial _\rho } + \tilde{d}^0_{\mathcal {S}}(b,\rho ), \end{aligned}$$

(5.26)

where

$$\begin{aligned}&\tilde{d}^\pm _{\mathcal {S}}(b,\rho ) = -2 e^{\pi b \rho } e^{\mp \pi b \left( \tfrac{i b}{2}+2 \theta _0+\theta _t \right) } \cosh \left( \pi b \left( \tfrac{i b}{2}+\theta _t \pm \rho \right) \right) , \end{aligned}$$

(5.27)

$$\begin{aligned}&\tilde{d}^0_{\mathcal {S}}(b,\rho ) = - 2 \cosh \left( 2 \pi b \left( \tfrac{i b}{2}+\theta _0+\theta _t\right) \right) - \tilde{d}^+_{\mathcal {S}}(b,\rho )-\tilde{d}^-_{\mathcal {S}}(b,\rho ). \end{aligned}$$

(5.28)

Proposition 5.4

For $(\sigma _s,\rho )\in \mathbb C^2$, the function $\mathcal {S}$ satisfies the following pair of difference equations:

$$\begin{aligned}&\tilde{D}_{\mathcal {S}}(b,\rho )\; \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ) = 2{\text {cosh}}{(2\pi b \sigma _s)}\; \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ), \end{aligned}$$

(5.29a)

$$\begin{aligned}&\tilde{D}_{\mathcal {S}}(b^{-1},\rho )\;\mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ) = 2{\text {cosh}}{(2\pi b^{-1} \sigma _s)}\; \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ). \end{aligned}$$

(5.29b)

5.3 Polynomial Limits

Our next two theorems state that $\mathcal {S}$ reduces to the Al-Salam Chihara polynomials when the variable $\rho$ is suitably discretized and to the little q-Jacobi polynomials when $\sigma _s$ is suitably discretized. The proofs proceed along the same lines as the proofs of Theorem 4.7 and Theorem 4.8 and are therefore omitted.

We make the following assumption which ensures that the poles of the integrand $I_\mathcal {S}$ in (5.6) are simple.

Assumption 5.5

Assume that $b>0$ is such that $b^2$ is irrational. Moreover, assume that

$$\begin{aligned} \theta _t, \text {Re} \,\sigma _s \ne 0, \qquad \text {Re} \,\big (\pm \sigma _s+ \rho +\theta _0\big ) \ne 0. \end{aligned}$$

(5.30)

Theorem 5.6

(From $\mathcal {S}$ to the Al-Salam Chihara polynomials) Let $\sigma _s \in \mathbb {C} \backslash \Delta _{\mathcal S,\sigma _s}$ and suppose that Assumptions 1.1 and 5.5 are satisfied. Define $\{\rho _n\}_{n=0}^\infty \subset \mathbb {C}$ by

$$\begin{aligned} \rho _n = \theta _t+\tfrac{i Q}{2}+inb. \end{aligned}$$

(5.31)

Under the parameter correspondence

$$\begin{aligned} \alpha _{S}=e^{2\pi b(\theta _t+\theta _0+\frac{iQ}{2})}, \qquad \beta _{S}= e^{2\pi b(\theta _t-\theta _0+\frac{iQ}{2})}, \qquad q=e^{2i\pi b^2}, \end{aligned}$$

(5.32)

the function $\mathcal {S}$ satisfies, for each $n\ge 0$,

$$\begin{aligned} \lim \limits _{\rho \rightarrow \rho _n} \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ) = S_n\left( e^{2\pi b \sigma _s};\alpha _{S},\beta _{S};q\right) , \end{aligned}$$

(5.33)

where $S_n$ are the Al-Salam Chihara polynomials defined in (B.17).

Theorem 5.7

(From $\mathcal {S}$ to the little q-Jacobi polynomials) Let $\rho \in \mathbb {C}\backslash \Delta _\rho$ and suppose that Assumptions 1.1 and 5.5 are satisfied. Under the parameter correspondence

$$\begin{aligned} \alpha _Y = e^{4\pi b \theta _t}, \qquad \beta _Y = e^{4\pi b \theta _0}, \qquad x_Y = e^{\pi b(iQ+2\theta _t)} e^{-2\pi b \rho }, \qquad q= e^{2i\pi b^2},\nonumber \\ \end{aligned}$$

(5.34)

the function $\mathcal {S}$ satisfies, for each $n\ge 0$,

$$\begin{aligned} \lim \limits _{\sigma _s\rightarrow \sigma _s^{(n)}} \mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho ) = Y_n \left( x_Y;\alpha _Y,\beta _Y,q \right) , \end{aligned}$$

(5.35)

where $\sigma _s^{(n)}$ is defined in (3.11) and where $Y_n$ are the little q-Jacobi polynomials defined in (B.24).

6 The Function $\mathcal {X}$

In this section, we define the function $\mathcal {X}(b,\theta ,\sigma _s,\omega )$ which generalizes continuous big q-Hermite polynomials. It lies at the fourth level of the non-polynomial scheme and is defined as a confluent limit of $\mathcal {S}$. We show that $\mathcal {X}$ is a joint eigenfunction of four difference operators and that it reduces to the continuous big q-Hermite polynomials, which lie at the fourth level of the q-Askey scheme, when $\omega$ is suitably discretized.

6.1 Definition and Integral Representation

Let $\theta$ and $\omega$ be two new parameters defined by

$$\begin{aligned} \theta _0 = \frac{\theta +\Lambda }{2}, \qquad \theta _t = \frac{\theta -\Lambda }{2},\qquad \rho = -\frac{\Lambda }{2}+\omega . \end{aligned}$$

(6.1)

Definition 6.1

The function $\mathcal {X}$ is defined by

$$\begin{aligned} \mathcal {X}(b,\theta ,\sigma _s,\omega ) = \lim \limits _{\Lambda \rightarrow +\infty } \mathcal {S}\left( b,\tfrac{\theta +\Lambda }{2}, \tfrac{\theta -\Lambda }{2},\sigma _s,-\tfrac{\Lambda }{2}+\omega \right) , \end{aligned}$$

(6.2)

where $\mathcal {S}$ is given in (5.4).

The next theorem shows that for each choice of $(b,\theta ) \in (0,\infty ) \times \mathbb R$, $\mathcal {X}$ is a well-defined analytic function of $(\sigma _s,\omega ) \in (\mathbb {C} \backslash \Delta _\mathcal {X,\sigma _s}) \times (\mathbb {C} \backslash \Delta _\omega )$, where $\Delta _\mathcal {S,\sigma _s}, \Delta _\omega \subset \mathbb C$ are discrete sets of points where $\mathcal S$ may have poles. The proof is omitted since it involves computations which are similar to those presented in the proofs of Theorems 4.2 and 5.2.

Theorem 6.2

Suppose that Assumption 1.1 is satisfied. The limit (6.2) exists uniformly for $(\sigma _s,\omega )$ in compact subsets of

$$\begin{aligned} \Omega _{\mathcal {X}}:= (\mathbb C \backslash \Delta _\mathcal {X,\sigma _s}) \times (\mathbb {C} \backslash \Delta _\omega ), \end{aligned}$$

(6.3)

where

$$\begin{aligned}&\Delta _\mathcal {X,\sigma _s} := \{\pm \sigma _s \, |\, \sigma _s \in \Delta _{\mathcal {X}, \sigma _s}'\}, \\&\Delta _\omega := \{\tfrac{\theta }{2}+\tfrac{iQ}{2}+ibm+ilb^{-1} \}_{m,l=0}^\infty \cup \{\tfrac{\theta }{2}-\tfrac{iQ}{2}-ibm-ilb^{-1} \}_{m,l=0}^\infty , \end{aligned}$$

with

$$\begin{aligned}&\Delta _\mathcal {X,\sigma _s}' := \{\theta + \tfrac{iQ}{2} + i m b +il b^{-1}\}_{m,l=0}^\infty \cup \{\theta - \tfrac{iQ}{2} - i m b - il b^{-1}\}_{m,l=0}^\infty . \end{aligned}$$

Moreover, the function $\mathcal {X}$ admits the following integral representation:

$$\begin{aligned} \mathcal {X}(b,\theta ,\sigma _s,\omega ) = P_\mathcal {X}(\sigma _s,\omega ) \displaystyle \int _{\mathcal {C}_{\mathcal {X}}} dx \; I_\mathcal {X}(x,\sigma _s,\omega ) \qquad \text {for }(\sigma _s,\omega ) \in \Omega _{\mathcal {X}}, \end{aligned}$$

(6.4)

where

$$\begin{aligned}&P_\mathcal {X}(\sigma _s,\omega ) = s_b\left( \omega -\tfrac{\theta }{2}\right) \prod _{\epsilon =\pm 1} s_b(\epsilon \sigma _s-\theta ), \end{aligned}$$

(6.5)

$$\begin{aligned}&I_\mathcal {X}(x,\sigma _s,\omega ) = e^{-i \pi x^2} e^{-i \pi x \left( \frac{5\theta }{2}+\frac{3iQ}{2}-\omega \right) } \frac{s_b\left( x+\frac{\theta }{2}-\omega \right) }{s_b(x+\frac{i Q}{2})} \prod _{\epsilon =\pm 1} s_b\left( x+\theta +\epsilon \sigma _s\right) , \end{aligned}$$

(6.6)

and the contour $\mathcal {C}_{\mathcal {X}}$ is any curve from $-\infty$ to $+\infty$ which separates the three decreasing from the increasing sequences of poles, with the requirement that its right tail satisfies

$$\begin{aligned} \text {Im} \,x + \frac{Q}{4} - \frac{\text {Im} \,\omega }{2} < - \delta \qquad \text {for }x \in \mathcal {C}_{\mathcal {X}}\text { with }\text {Re} \,x\text { sufficiently large}, \end{aligned}$$

(6.7)

for some $\delta >0$. In particular, $\mathcal X$ is a meromorphic function of $(\sigma _s,\omega ) \in \mathbb C^2$. If $(\sigma _s,\omega ) \in \mathbb {R}^2$, then the contour $\mathcal {C}_{\mathcal {X}}$ can be any curve from $-\infty$ to $+\infty$ lying within the strip $\text {Im} \,x \in (-Q/2,-Q/4-\delta )$.

Furthermore, as a consequence of (2.7), $\mathcal {X}$ satisfies

$$\begin{aligned} \mathcal {X}(b,\theta ,\sigma _s,\omega ) = \mathcal {X}(b^{-1},\theta ,\sigma _s,\omega ). \end{aligned}$$

(6.8)

6.2 Difference Equations

The function $\mathcal {X}(b,\theta ,\sigma _s,\omega )$ is a joint eigenfunction of four difference operators, two acting on $\sigma _s$ and the remaining two on $\omega$. This follows by taking the confluent limit (6.2) of the difference equations (5.3) and (5.4) satisfied by $\mathcal {S}$. The proofs of the following two propositions are omitted since they are similar to those presented in Sect. 4.2.

6.2.1 First Pair of Equations

Introduce a difference operator $D_{\mathcal {X}}(b,\sigma _s)$ such that

$$\begin{aligned} D_{\mathcal {X}}(b,\sigma _s) = d_{\mathcal {X}}^+(b,\sigma _s) e^{ib \partial _{\sigma _s}} + d_{\mathcal {X}}^+(b,-\sigma _s) e^{-ib \partial _{\sigma _s}} + d_{\mathcal {X}}^0(b,\sigma _s), \end{aligned}$$

(6.9)

where

$$\begin{aligned}&d_{\mathcal {X}}^+(b,\sigma _s) = -e^{-\frac{3}{2} \pi b (2 \sigma _s+i b)} \frac{\cosh \left( \pi b \left( \frac{i b}{2}+\theta +\sigma _s\right) \right) }{2 \sinh (2 \pi b \sigma _s) \sinh (\pi b (2 \sigma _s+i b))}, \end{aligned}$$

(6.10)

$$\begin{aligned}&d_{\mathcal {X}}^0(b,\sigma _s) = e^{-\pi b (\theta + i Q)} - d_{\mathcal {X}}^+(b,\sigma _s)-d_{\mathcal {X}}^+(b,-\sigma _s). \end{aligned}$$

(6.11)

Proposition 6.3

For $(\sigma _s,\omega ) \in \mathbb C^2$, the function $\mathcal {X}$ satisfies the following pair of difference equations:

$$\begin{aligned}&D_{\mathcal {X}}(b,\sigma _s) \; \mathcal {X}(b,\theta ,\sigma _s,\omega ) = e^{-2\pi b \omega } \; \mathcal {X}(b,\theta ,\sigma _s,\omega ) \end{aligned}$$

(6.12a)

$$\begin{aligned}&D_{\mathcal {X}}(b^{-1},\sigma _s) \; \mathcal {X}(b,\theta ,\sigma _s,\omega ) = e^{-2\pi b^{-1} \omega } \; \mathcal {X}(b,\theta ,\sigma _s,\omega ). \end{aligned}$$

(6.12b)

6.2.2 Second Pair of Equations

Introduce the difference operator $\tilde{D}_{\mathcal {X}}(b,\omega )$ such that

$$\begin{aligned} \tilde{D}_{\mathcal {X}}(b,\omega ) = \tilde{d}_{\mathcal {X}}^+(b,\omega ) e^{ib \partial _\omega } + \tilde{d}_{\mathcal {X}}^-(b,\omega ) e^{-ib \partial _\omega } + \tilde{d}_{\mathcal {X}}^0(b,\omega ), \end{aligned}$$

(6.13)

where

$$\begin{aligned}&\tilde{d}_{\mathcal {X}}^+(b,\omega ) = e^{-2\pi b(\theta +\frac{iQ}{2})}, \qquad \tilde{d}_{\mathcal {X}}^-(b,\omega ) = -2 e^{\pi b \big (\tfrac{i b}{2}+\frac{3 \theta }{2}+\omega \big )} \cosh \left( \pi b \left( \tfrac{i b}{2}+\tfrac{\theta }{2}-\omega \right) \right) , \\&\tilde{d}_{\mathcal {X}}^0(b,\omega ) = -2 \cosh (\pi b (2 \theta +i b)) - \tilde{d}_{\mathcal {X}}^+(b,\omega ) - \tilde{d}_{\mathcal {X}}^-(b,\omega )=e^{\pi b (\theta +2 \omega )}. \end{aligned}$$

Proposition 6.4

For $(\sigma _s,\omega ) \in \mathbb C^2$, the function $\mathcal {X}$ satisfies the following pair of difference equations:

$$\begin{aligned}&\tilde{D}_{\mathcal {X}}(b,\omega ) \; \mathcal {X}(b,\theta ,\sigma _s,\omega ) = 2{\text {cosh}}{(2\pi b \sigma _s)} \; \mathcal {X}(b,\theta ,\sigma _s,\omega ) , \end{aligned}$$

(6.14a)

$$\begin{aligned}&\tilde{D}_{\mathcal {X}}(b^{-1},\omega ) \; \mathcal {X}(b,\theta ,\sigma _s,\omega ) = 2{\text {cosh}}{(2\pi b^{-1} \sigma _s)} \; \mathcal {X}(b,\theta ,\sigma _s,\omega ). \end{aligned}$$

(6.14b)

6.3 Polynomial Limit

Our next theorem shows that $\mathcal {X}$ reduces to the continuous big q-Hermite polynomials when $\omega$ is suitably discretized. We make the following assumption, which implies that all the poles of the integrand $I_\mathcal {X}$ are simple and that $q=e^{2i\pi b^2}$ is not a root of unity.

Assumption 6.5

Assume that $b>0$ is such that $b^2$ is irrational and that

$$\begin{aligned} \text {Re} \,\sigma _s \ne 0, \qquad \text {Re} \,\big (\omega +\tfrac{\theta }{2}\pm \sigma _s\big ) \ne 0. \end{aligned}$$

(6.15)

The proof of the next theorem is analogous to the proof of Theorem 4.7 and is omitted.

Theorem 6.6

(From $\mathcal {X}$ to the continuous big q-Hermite polynomials) Let $\sigma _s \in \mathbb C \backslash \Delta _\mathcal {X,\sigma _s}$ and suppose that Assumptions 1.1 and 6.5 are satisfied. Define $\{\omega _n\}_{n=0}^\infty \subset \mathbb {C}$ by

$$\begin{aligned} \omega _n = \frac{\theta }{2}+\frac{iQ}{2}+ibn. \end{aligned}$$

(6.16)

Under the parameter correspondence

$$\begin{aligned} \alpha _{X} = e^{2\pi b (\theta +\frac{i Q}{2})}, \qquad q=e^{2i\pi b^2}, \end{aligned}$$

(6.17)

the function $\mathcal {X}$ satisfies, for each $n\ge 0$,

$$\begin{aligned} \lim \limits _{\omega \rightarrow \omega _n} \mathcal {X}(b,\theta ,\omega ,\sigma _s) = X_n(e^{2\pi b \sigma _s};\alpha _{X},q), \end{aligned}$$

(6.18)

where $X_n$ are the continuous big q-Hermite polynomials defined in (B.37).

7 The Function $\mathcal {Q}$

The function $\mathcal {Q}(b,\sigma _s,\eta )$ is defined as a confluent limit of $\mathcal {X}$ and is one of the two elements at the fifth and lowest level of the non-polynomial scheme. We show that $\mathcal {Q}$ is a joint eigenfunction of four difference operators, two acting on $\sigma _s$ and two acting on $\eta$. Finally, we show that $\mathcal {Q}$ reduces to the continuous q-Hermite polynomials, which lie at the lowest level of the q-Askey scheme, when $\eta$ is suitably discretized.

Interestingly, the mechanism behind the polynomial limit for $\mathcal {Q}$ is different from that of all the other polynomial limits in this paper. In all other cases, the simple pole of the contour integral which compensates for the simple zero in the prefactor arises because the contour of integration is squeezed between two poles of the integrand. However, in the case of $\mathcal {Q}$, the simple pole of the contour integral arises because the integrand loses its decay at infinity in the relevant polynomial limit.

The derivation of the difference equations for $\mathcal {Q}$ also presents some novelties compared to the other derivations of difference equations appearing in this paper. More precisely, the first pair of difference equations for $\mathcal {Q}$ obtained by taking the confluent limit of the difference equations for $\mathcal {X}$ are “squares” of the simplest possible difference equations for $\mathcal {Q}$. By finding square roots of the relevant difference operators, we obtain the equations in their simplified form (this is the form that reduces to the standard difference equations for the continuous q-Hermite polynomials in the polynomial limit).

7.1 Definition and Integral Representation

Let $\eta$ be a new parameter defined as follows:

$$\begin{aligned} \omega = \frac{\theta }{2}+\eta . \end{aligned}$$

(7.1)

Moreover, define the normalization factor K by

$$\begin{aligned} K(\eta ,\theta ) = e^{2 i \pi \left( \eta - \frac{i Q}{2}\right) \left( \theta +\frac{i Q}{2}\right) }. \end{aligned}$$

(7.2)

The factor K is a non-polynomial analog of the factor $\alpha ^{-n}$ in (B.40).

Definition 7.1

The function $\mathcal {Q}$ is defined by

$$\begin{aligned} \mathcal {Q}(b,\sigma _s,\eta ) = \lim \limits _{\theta \rightarrow -\infty } \left( K(\eta ,\theta ) \mathcal {X}(b,\theta ,\sigma _s,\tfrac{\theta }{2}+\eta )\right) . \end{aligned}$$

(7.3)

The next theorem shows that $\mathcal Q$ is a well-defined and analytic function of $(\sigma _s,\eta ) \in \mathbb C \times (\{\text {Im} \,\eta < Q/2 \} \backslash \Delta _\eta )$, where $\Delta _\eta$ is a discrete set of points at which $\mathcal Q$ may have poles. The theorem also provides an integral representation for $\mathcal Q$. Even if the requirement $\text {Im} \,\eta < Q/2$ is needed to ensure that the integral in the representation for $\mathcal {Q}$ converges, it will follow from the difference equations established later that $\mathcal Q$ extends to a meromorphic function of $\eta \in \mathbb {C}$.

Theorem 7.2

Suppose that Assumption 1.1 is satisfied. The limit (7.3) exists uniformly for $(\sigma _s,\eta )$ in compact subsets of

$$\begin{aligned} \Omega _{\mathcal {Q}}:= \mathbb C \times (\{ \text {Im} \,\eta <Q/2 \} \backslash \Delta _\eta ), \end{aligned}$$

(7.4)

where

$$\begin{aligned} \Delta _\eta := \{-\tfrac{iQ}{2} -imb - ilb^{-1} \}_{m,l=0}^\infty . \end{aligned}$$

(7.5)

Moreover, the function $\mathcal {Q}$ admits the following integral representation:

$$\begin{aligned} \mathcal {Q}(b,\sigma _s,\eta ) = P_\mathcal {Q}(\sigma _s,\eta ) \displaystyle \int _{\mathcal {C}_{\mathcal {Q}}} dx\; I_\mathcal {Q}(x,\sigma _s,\eta ) \qquad \text {for }(\sigma _s,\eta ) \in \Omega _{\mathcal {Q}}, \end{aligned}$$

(7.6)

where

$$\begin{aligned}&P_\mathcal {Q}(\sigma _s,\eta ) = e^{i \pi \left( \frac{1}{6}+\frac{7 Q^2}{24}-\frac{\eta ^2}{2}+i \eta Q- \sigma _s^2\right) } s_b(\eta ), \end{aligned}$$

(7.7)

$$\begin{aligned}&I_\mathcal {Q}(x,\sigma _s,\eta ) = e^{-i\pi x^2} e^{2i\pi x(\eta -\frac{iQ}{2})} s_b\left( x+\sigma _s\right) s_b\left( x-\sigma _s\right) , \end{aligned}$$

(7.8)

and the contour of integration $\mathcal {C}_{\mathcal {Q}}$ is any curve from $-\infty$ to $+\infty$ passing above the points $x= \pm \sigma _s - iQ/2$ and with the requirement that its right tail satisfies

$$\begin{aligned} \text {Im} \,x + \tfrac{Q}{4} - \tfrac{ \text {Im} \,\eta }{2} < -\delta \qquad \text {for all }x \in \mathcal {C}_{\mathcal {Q}}\text { with }\text {Re} \,x\text { sufficiently large}, \end{aligned}$$

(7.9)

for some $\delta >0$. If $(\sigma _s,\eta ) \in \mathbb R^2$, then the contour $\mathcal {C}_{\mathcal {Q}}$ can be any curve from $-\infty$ to $+\infty$ lying within the strip $\text {Im} \,x \in (-Q/2,-Q/4-\delta )$.

Proof

The proof will be omitted since it involves computations which are similar to those presented in the proofs of Theorems 4.2 and  5.2. However, we point out that before taking the limit $\theta \rightarrow -\infty$ of the integral representation (6.4) for $\mathcal {X}$, one should make the change of variables $x \rightarrow x - \theta$ and thus write

$$\begin{aligned} K(\eta , \theta ) P_\mathcal {X}(\sigma _s,\tfrac{\theta }{2}+\eta ) I_\mathcal {X}(x-\theta ,\sigma _s,\tfrac{\theta }{2}+\eta ) = P_\mathcal {Q}(\sigma _s,\eta ) Z_\mathcal {Q}(x, \theta ) I_\mathcal {Q}(x,\sigma _s,\eta ), \nonumber \\\end{aligned}$$

(7.10)

where

$$\begin{aligned} Z_\mathcal {Q}(x, \theta ){} & {} := e^{\frac{i \pi }{6} (3 \eta ^2+\frac{5 Q^2}{4}+6 \sigma _s^2-1)} e^{\frac{\pi Q}{2}(x-\theta )} e^{-i \pi \eta x}\nonumber \\{} & {} \quad \times \ e^{i \pi \theta (\eta +\theta )} \frac{s_b(x-\eta -\theta )s_b(\sigma _s-\theta )s_b(-\sigma _s-\theta ) }{s_b(x-\theta +\frac{i Q}{2})} \end{aligned}$$

(7.11)

satisfies $Z_\mathcal {Q}(x, \theta ) \rightarrow 1$ as $\theta \rightarrow -\infty$. $\square$

Thanks to (2.7), we have

$$\begin{aligned} \mathcal {Q}(b,\sigma _s,\eta ) = \mathcal {Q}(b^{-1},\sigma _s,\eta ). \end{aligned}$$

(7.12)

Remark 7.3

A close relative of the function $\mathcal {Q}$ has appeared in [24, 54] in the context of quantum relativistic Toda systems. More precisely, the function $\mathcal {H}(a_-,a_+,x,y)$ in [54, Eq. (5.56)] is related to $\mathcal {Q}$ by

$$\begin{aligned} \mathcal {Q}(b,\sigma _s,\eta ) = \; e^{\frac{1}{4} i \pi \big (\frac{Q^2}{4}+\frac{1}{2}-\eta ^2-8 \sigma _s^2\big )} g_b(\eta ) \; \mathcal {H}(b,b^{-1},-\eta ,2\sigma _s), \end{aligned}$$

(7.13)

where $g_b(z)$ satisfies $s_b(z)=g_b(z)/g_b(-z)$ and is defined by

$$\begin{aligned} g_b(z){} & {} ={\text {exp}}{\left\{ \int _0^\infty \frac{dt}{t}\left[ \frac{e^{2i z t}-1}{4 {\text {sinh}}{b t} {\text {sinh}}{b^{-1} t}}+\frac{1}{4}z^2 \left( e^{-2bt}+e^{-\frac{2t}{b}}\right) -\frac{iz}{2t} \right] \right\} },\nonumber \\{} & {} \qquad \text {for } \text {Im} \,z > -\frac{Q}{2}. \end{aligned}$$

(7.14)

7.2 Difference Equations

We show that $\mathcal {Q}(b,\sigma _s,\eta )$ is a joint eigenfunction of four difference operators, two acting on $\sigma _s$ and the remaining two on $\eta$.

We know from Theorem 7.2 that $\mathcal Q$ is a well-defined analytic function of $\sigma _s \in \mathbb C$ and a meromorphic function of $\eta$ for $\text {Im} \,\eta <Q/2$. The difference equations will first be derived as equalities between meromorphic functions defined on this limited domain. However, the difference equations in $\eta$ can then be used to extend the results to $\eta \in \mathbb {C}$, see Proposition 7.8.

7.2.1 First Pair of Equations

Define the difference operator $D_\mathcal {Q}(b,\sigma _s)$ by

$$\begin{aligned} D_\mathcal {Q}(b,\sigma _s) = d^+_\mathcal {Q}(b,\sigma _s) e^{ib \partial _{\sigma _s}} + d^+_\mathcal {Q}(b,-\sigma _s) e^{-ib \partial _{\sigma _s}} + d^0_\mathcal {Q}(b,\sigma _s), \end{aligned}$$

(7.15)

where

$$\begin{aligned}&d^+_\mathcal {Q}(b,\sigma _s) = -\frac{e^{-2 \pi b (2 \sigma _s+i b)}}{4 \sinh (2 \pi b \sigma _s) \sinh (\pi b (2 \sigma _s+i b))}, \end{aligned}$$

(7.16)

$$\begin{aligned}&d^0_\mathcal {Q}(b,\sigma _s) = e^{-i\pi b Q} - d^+_\mathcal {Q}(b,\sigma _s) - d^+_\mathcal {Q}(b,-\sigma _s). \end{aligned}$$

(7.17)

Proposition 7.4

For $\sigma _s \in \mathbb C$ and $\text {Im} \,\eta < Q/2$, the function $\mathcal {Q}$ satisfies the following pair of difference equations:

$$\begin{aligned}&D_\mathcal {Q}(b,\sigma _s) \; \mathcal {Q}(b,\sigma _s,\eta ) = e^{-2\pi b \eta } \; \mathcal {Q}(b,\sigma _s,\eta ), \end{aligned}$$

(7.18a)

$$\begin{aligned}&D_\mathcal {Q}(b^{-1},\sigma _s) \; \mathcal {Q}(b,\sigma _s,\eta ) = e^{-2\pi b^{-1} \eta } \; \mathcal {Q}(b,\sigma _s,\eta ). \end{aligned}$$

(7.18b)

Proof

It is straightforward to verify that the following limits hold:

$$\begin{aligned} \lim \limits _{\theta \rightarrow -\infty } e^{\pi b \theta } d^+_\mathcal {X}(b,\pm \sigma _s) = d^+_\mathcal {Q}(b,\pm \sigma _s), \qquad \lim \limits _{\theta \rightarrow -\infty } e^{\pi b \theta } d^0_\mathcal {X}(b,\sigma _s) = d^0_\mathcal {Q}(b,\sigma _s), \nonumber \\\end{aligned}$$

(7.19)

where $d^+_\mathcal {X}$ and $d^0_\mathcal {X}$ are defined in (6.10) and (6.11), respectively. It follows that

$$\begin{aligned} \lim \limits _{\theta \rightarrow -\infty } e^{\pi b \theta } D_\mathcal {X}(b,\sigma _s) = D_\mathcal {Q}(b,\sigma _s). \end{aligned}$$

(7.20)

where $D_\mathcal {X}$ is defined in (6.9). The difference equation (7.18a) follows after taking the confluent limit (7.3) of the difference equation (6.12a) and utilizing (7.20). $\square$

In what follows, we show that the function $\mathcal {Q}$ satisfies a pair of difference equations which is more fundamental than (7.4).

Proposition 7.5

Define the difference operator $\hat{D}_\mathcal {Q}(b,\sigma _s)$ by

$$\begin{aligned} \hat{D}_\mathcal {Q}(b,\sigma _s) = \hat{d}^+_\mathcal {Q}(b,\sigma _s) e^{\frac{ib}{2}\partial _{\sigma _s}} + \hat{d}^-_\mathcal {Q}(b,\sigma _s) e^{-\frac{ib}{2}\partial _{\sigma _s}}, \end{aligned}$$

(7.21)

with

$$\begin{aligned} \hat{d}^+_\mathcal {Q}(b,\sigma _s)=-\frac{e^{-\pi b \left( 2\sigma _s+\frac{i Q}{2}\right) }}{2 \sinh (2\pi b \sigma _s)}, \qquad \hat{d}^-_\mathcal {Q}(b,\sigma _s)=\hat{d}^+_\mathcal {Q}(b,-\sigma _s). \end{aligned}$$

(7.22)

The following operator identity holds:

$$\begin{aligned} \left( \hat{D}_\mathcal {Q}(b,\sigma _s)\right) ^2 = D_\mathcal {Q}(b,\sigma _s), \end{aligned}$$

(7.23)

where $D_\mathcal {Q}$ is defined in (7.15).

Proof

The left-hand side of (7.23) can be written as

$$\begin{aligned} \left( \hat{D}_\mathcal {Q}(b,\sigma _s)\right) ^2 =&\; \hat{d}^+_\mathcal {Q}(b,\sigma _s) \hat{d}^+_\mathcal {Q}(b,\sigma _s+\tfrac{ib}{2}) e^{ib\partial _{\sigma _s}} + \hat{d}^-_\mathcal {Q}(b, \sigma _s) \hat{d}^-_\mathcal {Q}(b,\sigma _s-\tfrac{ib}{2}) e^{-ib\partial _{\sigma _s}} \nonumber \\&+ \hat{d}^+_\mathcal {Q}(b,\sigma _s) \hat{d}^-_\mathcal {Q}(b,\sigma _s+\tfrac{ib}{2}) + \hat{d}^-_\mathcal {Q}(b,\sigma _s) \hat{d}^+_\mathcal {Q}(b,\sigma _s-\tfrac{ib}{2}) \end{aligned}$$

(7.24)

and straightforward computations show that the right-hand side coincides with the operator $D_\mathcal {Q}(b,\sigma _s)$. $\square$

We next show that the difference equations (7.4) satisfied by the function $\mathcal {Q}$ can be simplified using the identity (7.23). The simplified equations can be viewed as “square roots” of the equations in (7.4).

Proposition 7.6

For $\sigma _s \in \mathbb C$ and $\text {Im} \,\eta < Q/2$, the function $\mathcal {Q}$ satisfies

$$\begin{aligned}&\hat{D}_\mathcal {Q}(b,\sigma _s) \; \mathcal {Q}(b,\sigma _s,\eta ) = e^{-\pi b \eta } \; \mathcal {Q}(b,\sigma _s,\eta ), \end{aligned}$$

(7.25a)

$$\begin{aligned}&\hat{D}_\mathcal {Q}(b^{-1},\sigma _s) \; \mathcal {Q}(b,\sigma _s,\eta ) = e^{-\pi b^{-1} \eta } \; \mathcal {Q}(b,\sigma _s,\eta ). \end{aligned}$$

(7.25b)

Proof

Let us rewrite (7.6) as

$$\begin{aligned} \mathcal {Q}(b,\sigma _s,\eta ) = \displaystyle \int _{\mathcal {C}_{\mathcal {Q}}} dx \; \psi (x,\sigma _s,\eta ) e^{2i\pi x \eta }, \end{aligned}$$

(7.26)

where

$$\begin{aligned} \psi (x,\sigma _s,\eta ) = e^{i \pi \left( \frac{1}{6}+\frac{7 Q^2}{24}-\frac{\eta ^2}{2}+i \eta Q-\sigma _s^2\right) } \; s_b(\eta ) e^{-i\pi x^2} e^{\pi Q x} s_b\left( x+\sigma _s\right) s_b\left( x-\sigma _s\right) , \end{aligned}$$

(7.27)

and where the contour $\mathcal {C}_{\mathcal {Q}}$ is such that it passes above the points $x= \pm \sigma _s$, and such that $\text {Im} \,x < \tfrac{ \text {Im} \,\eta }{2} - Q/4 - \delta$ for some small but fixed $\delta > 0$ as $\text {Re} \,x \rightarrow +\infty$. The following estimates, which are easily established with the help of (2.8), imply that the integrand has exponential decay on $\mathcal {C}_{\mathcal {Q}}$ as $\text {Re} \,x \rightarrow +\infty$:

$$\begin{aligned} \psi (x,\sigma _s,\eta ) e^{2i\pi x \eta } = {\left\{ \begin{array}{ll} O\left( e^{4\pi (\text {Re} \,x)( \text {Im} \,x+\frac{Q}{4}-\frac{\text {Im} \,\eta }{2})} \right) , &{} \text {Re} \,x \rightarrow +\infty , \\ O\left( e^{2\pi \text {Re} \,x (\frac{Q}{2} - \text {Im} \,\eta )} \right) , &{} \text {Re} \,x \rightarrow -\infty , \end{array}\right. } \end{aligned}$$

(7.28)

uniformly for $(b,\text {Im} \,x,\sigma _s,\eta )$ in compact subsets of $(0,\infty ) \times \mathbb R \times \mathbb {C}^2$.

Utilizing the difference equation (2.5), we verify that the following identity holds:

$$\begin{aligned} \hat{D}_\mathcal {Q}(b,\sigma _s) \psi (x,\sigma _s,\eta ) = \psi (x-\tfrac{ib}{2},\sigma _s,\eta ). \end{aligned}$$

(7.29)

Letting the difference operator $\hat{D}_\mathcal {Q}$ act inside the contour integral in (7.26) and using (7.29), we obtain

$$\begin{aligned} \hat{D}_\mathcal {Q}(b,\sigma _s) \mathcal {Q}(b,\sigma _s,\eta ) = \displaystyle \int _{\mathcal {C}_{\mathcal {Q}}} dx \; \psi (x-\tfrac{ib}{2},\sigma _s,\eta ) e^{2i\pi x \eta }. \end{aligned}$$

Performing the change of variables $x = y + ib/2$, we find

$$\begin{aligned} \hat{D}_\mathcal {Q}(b,\sigma _s) \mathcal {Q}(b,\sigma _s,\eta ) = e^{-\pi b \eta } \int _{\mathcal {C}_{\mathcal {Q}} - \frac{ib}{2}} dy \; \psi (y,\sigma _s,\eta ) e^{2i\pi y \eta }. \end{aligned}$$

Deforming the contour back to $\mathcal {C}_{\mathcal {Q}}$, noting that no poles are crossed and that the integrand retains its exponential decay at infinity throughout the deformation, we arrive at (7.25a). The difference equation (7.25b) follows from (7.25a) and the symmetry (7.12) of $\mathcal {Q}$. $\square$

7.2.2 Second Pair of Equations

Define the difference operator $\tilde{D}_\mathcal {Q}(b,\eta )$ by

$$\begin{aligned} \tilde{D}_\mathcal {Q}(b,\eta ) = e^{ib\partial _\eta } + \left( 1+e^{2 \pi b \left( \eta -\frac{i b}{2}\right) } \right) e^{-ib\partial _\eta }. \end{aligned}$$

(7.30)

Proposition 7.7

For $\sigma _s \in \mathbb C$ and $\text {Im} \,(\eta +ib^{\pm 1}) < Q/2$, the function $\mathcal {Q}$ satisfies the pair of difference equations

$$\begin{aligned}&\tilde{D}_\mathcal {Q}(b,\eta ) \; \mathcal {Q}(b,\sigma _s,\eta ) = 2{\text {cosh}}{(2\pi b \sigma _s)} \; \mathcal {Q}(b,\sigma _s,\eta ), \end{aligned}$$

(7.31a)

$$\begin{aligned}&\tilde{D}_\mathcal {Q}(b^{-1},\eta ) \; \mathcal {Q}(b,\sigma _s,\eta ) = 2{\text {cosh}}{(2\pi b^{-1} \sigma _s)} \; \mathcal {Q}(b,\sigma _s,\eta ). \end{aligned}$$

(7.31b)

Proof

The difference equation (6.14a) can be rewritten as follows:

$$\begin{aligned}{} & {} \left( K(\eta ,\theta ) \tilde{D}_{\mathcal {X}}(b,\tfrac{\theta }{2}+\eta ) K(\eta ,\theta )^{-1} \right) K(\eta ,\theta ) \mathcal {X}(b,\theta ,\sigma _s,\tfrac{\theta }{2}+\eta ) \nonumber \\{} & {} \quad = 2{\text {cosh}}{(2\pi b \sigma _s)} \; K(\eta ,\theta ) \mathcal {X}(b,\theta ,\sigma _s,\tfrac{\theta }{2}+\eta ), \end{aligned}$$

(7.32)

where the operator $\tilde{D}_{\mathcal {X}}$ is given in (6.13). Moreover, we have

$$\begin{aligned} \lim \limits _{\theta \rightarrow -\infty } K(\eta ,\theta ) \tilde{D}_{\mathcal {X}}(b,\tfrac{\theta }{2}+\eta ) K(\eta ,\theta )^{-1} = \tilde{D}_\mathcal {Q}(b,\eta ). \end{aligned}$$

(7.33)

Taking the limit $\theta \rightarrow -\infty$ of (7.32) and recalling the definition (7.3), we obtain (7.31a). Finally, (7.31b) follows from (7.31a) and the symmetry (7.12) of $\mathcal {Q}$. $\square$

The next proposition shows that $\mathcal {Q}$ extends to a meromorphic function of $\eta$ everywhere in the complex plane. The proof will be omitted, since it is similar to that of Proposition 4.5.

Proposition 7.8

Let $b \in (0,\infty )$ and $\sigma _s \in \mathbb C$. Then there is a discrete subset $\Delta \subset \mathbb C$ such that the limit in (7.3) exists for all $\eta \in \mathbb C \backslash \Delta$. Moreover, the function $\mathcal Q$ defined by (7.3) is a meromorphic function of $(\sigma _s,\eta ) \in \mathbb C^2$ and the four difference equations (7.25) and (7.7) hold as equalities between meromorphic functions of $(\sigma _s,\eta ) \in \mathbb C^2$.

7.3 Polynomial Limit

In this subsection, we show that the function $\mathcal {Q}$ reduces to the continuous q-Hermite polynomials when $\eta$ is suitably discretized.

Theorem 7.9

(From $\mathcal {Q}$ to the continuous q-Hermite polynomials) Let $\sigma _s \in \mathbb {C}$. Suppose that $b>0$ is such that $b^2$ is irrational. Define $\{\eta _n\}_{n=0}^\infty \subset \mathbb {C}$ by

$$\begin{aligned} \eta _n = \tfrac{iQ}{2}+ibn. \end{aligned}$$

(7.34)

For each integer $n \ge 0$, the function $\mathcal {Q}$ satisfies

$$\begin{aligned} \lim \limits _{\eta \rightarrow \eta _n} \mathcal {Q}(b,\eta ,\sigma _s) = Q_n\left( e^{2\pi b \sigma _s};e^{2i\pi b^2}\right) , \end{aligned}$$

(7.35)

where $Q_n$ are the continuous q-Hermite polynomials defined in (B.40).

In order to prove Theorem 7.9, we will need the following two lemmas.

Lemma 7.10

For each $\epsilon >0$, the integrand $I_\mathcal {Q}$ defined in (7.8) obeys the estimates

$$\begin{aligned}&{\text {ln}}{\left( I_\mathcal {Q}(x,\sigma _s,\eta ) \right) } = \mathcal {A}^+_\mathcal {Q}(x,\eta ) + O\Big ( e^{-\frac{2\pi (1-\epsilon )}{\max (b,b^{-1})}|\text {Re} \,{x}|}\Big ), \qquad \text {Re} \,{x} \rightarrow +\infty , \end{aligned}$$

(7.36a)

$$\begin{aligned}&{\text {ln}}{\left( I_\mathcal {Q}(x,\sigma _s,\eta ) \right) } = \mathcal {A}^-_\mathcal {Q}(x,\eta ) + O\Big ( e^{-\frac{2\pi (1-\epsilon )}{\max (b,b^{-1})}|\text {Re} \,{x}|}\Big ), \qquad \text {Re} \,{x} \rightarrow -\infty , \end{aligned}$$

(7.36b)

uniformly for $(b,\text {Im} \,x, \sigma _s,\eta )$ in compact subsets of $(0,\infty ) \times \mathbb {R} \times \mathbb {C}^2$, where

$$\begin{aligned}&\mathcal {A}^+_\mathcal {Q}(x,\eta ) = -2i\pi x^2 + 2i\pi x(\eta -\tfrac{iQ}{2})-i\pi \left( \sigma _s^2 + \tfrac{Q^2-2}{12}\right) , \end{aligned}$$

(7.37a)

$$\begin{aligned}&\mathcal {A}^-_\mathcal {Q}(x,\eta ) = 2i\pi x(\eta -\tfrac{iQ}{2})+i\pi \left( \sigma _s^2 + \tfrac{Q^2-2}{12}\right) . \end{aligned}$$

(7.37b)

Proof

The lemma follows from (2.8) and (7.8). $\square$

Lemma 7.11

Let

$$\begin{aligned} \epsilon _\mathcal {Q}(x,\eta ) = I_\mathcal {Q}(x,\sigma _s,\eta ) - e^{\mathcal {A}_\mathcal {Q}^-(x,\eta )}, \end{aligned}$$

(7.38)

where $\mathcal {A}_\mathcal {Q}^-$ is defined in (7.37b). There is a neighborhood $U_0$ of $\eta _0=iQ/2$ and constants $c > 0$ and $M>0$ such that

$$\begin{aligned} \left|\epsilon _\mathcal {Q}(x,\eta ) \right|\le M e^{-c|\text {Re} \,x|} \qquad \text {for }\text {Re} \,x \le 0\text { and for }\eta \in U_0\text { with }\text {Im} \,\eta \le \text {Im} \,\eta _0, \nonumber \\\end{aligned}$$

(7.39)

uniformly for $(\text {Im} \,x, \sigma _s)$ in compact subsets of $\mathbb {R} \times \mathbb {C}$.

Proof

Utilizing the estimate (7.36b) with $\epsilon = 1/2$, we find that

$$\begin{aligned}{} & {} \epsilon _\mathcal {Q}(x,\eta ) = e^{\mathcal {A}_\mathcal {Q}^-(x,\eta )} \Big ( e^{O\left( \text {exp}(\frac{\pi \text {Re} \,x}{\max (b,b^{-1})}) \right) } - 1 \Big ) \nonumber \\{} & {} \quad = O\left( e^{\mathcal {A}_\mathcal {Q}^-(x,\eta )} e^{-\frac{\pi |\text {Re} \,x|}{\max (b,b^{-1})}} \right) , \qquad \text {Re} \,x \rightarrow -\infty , \end{aligned}$$

(7.40)

uniformly for $(b,\text {Im} \,x,\sigma _s,\eta )$ in compact subsets of $(0,\infty ) \times \mathbb {R}\times \mathbb {C}^2$. Since

$$\begin{aligned} e^{\text {Re} \,\mathcal {A}_\mathcal {Q}^-(x,\eta )} = e^{2\pi (\text {Re} \,x)(\frac{Q}{2}-\text {Im} \,\eta )} e^{-2\pi (\text {Im} \,x) (\text {Re} \,\eta )}e^{-2\pi (\text {Im} \,\sigma _s) (\text {Re} \,\sigma _s)}, \end{aligned}$$

(7.41)

the desired conclusion follows. $\square$

Proof of Theorem 7.9

Suppose $b >0$ and $\sigma _s \in \mathbb {C}$. The function $I_{\mathcal {Q}}(x,\sigma _s,\eta )$ has two downward sequences of poles starting at $x = \pm \sigma _s - iQ/2$. Consider the representation (7.6) for $\mathcal {Q}$ with the contour $\mathcal {C}_{\mathcal {Q}}$ passing above the points $x= \pm \sigma _s - iQ/2$ and satisfying (7.9) on its right tail.

Taking $\epsilon =1/2$ in the estimate (7.36a), we find that there exists a constant $M_1$ such that

$$\begin{aligned} \left|I_\mathcal {Q}(x,\sigma _s,\eta ) \right|\le M_1 \big | e^{\mathcal {A}_\mathcal {Q}^+(x,\eta )} \big | = M_1 e^{\text {Re} \,\mathcal {A}_\mathcal {Q}^+(x,\eta )}, \qquad x \in \mathcal {C}_{\mathcal {Q}}, \; \text {Re} \,x \ge 0, \nonumber \\\end{aligned}$$

(7.42)

uniformly for $(b,\sigma _s, \eta )$ in compact subsets of $(0,\infty ) \times \mathbb {C}^2$. Using (7.9) and noting that

$$\begin{aligned} e^{\text {Re} \,\mathcal {A}^+_\mathcal {Q}(x,\eta )} = e^{4\pi (\text {Re} \,x)\left( \text {Im} \,x+\frac{Q}{4} - \frac{\text {Im} \,\eta }{2} \right) } e^{-2\pi (\text {Im} \,x) (\text {Re} \,\eta )}e^{2\pi (\text {Im} \,\sigma _s)(\text {Re} \,\sigma _s)}, \end{aligned}$$

(7.43)

we infer the existence of a neighborhood $U_0$ of $\eta _0=iQ/2$ and constants $c > 0$ and $M>0$ such that

$$\begin{aligned} \left|I_\mathcal {Q}(x,\sigma _s,\eta ) \right|\le M e^{-c \text {Re} \,x} \end{aligned}$$

(7.44)

for all $x \in \mathcal {C}_{\mathcal {Q}}$ with $\text {Re} \,x \ge 0$ and for all $\eta \in U_0$.

Writing $\mathcal {C}_{\mathcal {Q}} = \mathcal {C}_{\mathcal {Q}}^+ \cup \mathcal {C}_{\mathcal {Q}}^-$, where $\mathcal {C}_{\mathcal {Q}}^+ = \mathcal {C}_{\mathcal {Q}} \cap \{\text {Re} \,x \ge 0\}$ and $\mathcal {C}_{\mathcal {Q}}^- = \mathcal {C}_{\mathcal {Q}} \cap \{\text {Re} \,x \le 0\}$, we can express (7.6) as

$$\begin{aligned}{} & {} \mathcal {Q}(b,\sigma _s,\eta )\nonumber \\{} & {} \quad = P_\mathcal {Q}(\sigma _s,\eta ) \displaystyle \int _{\mathcal {C}_{\mathcal {Q}}^+} dx\; I_\mathcal {Q}(x,\sigma _s,\eta ) + P_\mathcal {Q}(\sigma _s,\eta ) \displaystyle \int _{\mathcal {C}_{\mathcal {Q}}^-} dx\; I_\mathcal {Q}(x,\sigma _s,\eta ).\nonumber \\ \end{aligned}$$

(7.45)

The prefactor $P_\mathcal {Q}$ defined in (7.7) has a simple zero at $\eta = \eta _0=iQ/2$. Hence, by (7.44), the first term on the right-hand side of (7.45) vanishes in the limit $\eta \rightarrow \eta _0$. On the other hand, by (7.38),

$$\begin{aligned} \displaystyle \int _{\mathcal {C}_{\mathcal {Q}}^-} dx\; I_\mathcal {Q}(x,\sigma _s,\eta ) = \displaystyle \int _{\mathcal {C}_{\mathcal {Q}}^-} dx\; e^{\mathcal {A}_\mathcal {Q}^-(x,\eta )} + \displaystyle \int _{\mathcal {C}_{\mathcal {Q}}^-} dx \; \epsilon _\mathcal {Q}(x,\eta ). \end{aligned}$$

(7.46)

In view of Lemma 7.11, there exist constants $M_3, M_4 > 0$ such that

$$\begin{aligned} \left|\displaystyle \int _{\mathcal {C}_{\mathcal {Q}}^-} dx \; \epsilon _\mathcal {Q}(x,\eta ) \right|\le M_3 \displaystyle \int _{\mathcal {C}_{\mathcal {Q}}^-} dx \; e^{-c|\text {Re} \,x|} \le M_4 \end{aligned}$$

(7.47)

for all $\eta$ in a small neighborhood of $\eta _0$ with $\text {Im} \,\eta \le \text {Im} \,\eta _0$.

By Proposition 7.8, $\mathcal Q$ is a meromorphic function of $\eta \in \mathbb {C}$, and so has at most a pole at $\eta _0$. To prove (7.35), it is therefore enough to consider the limit $\eta \rightarrow \eta _0$ with $\eta$ such that $\text {Im} \,\eta < \text {Im} \,\eta _0 = Q/2$. In the remainder of the proof, we assume $\text {Im} \,\eta < Q/2$. Then, after multiplication by the prefactor $P_\mathcal {Q}$, the second term on the right-hand side of (7.46) vanishes in the limit $\eta \rightarrow \eta _0$ as a consequence of (7.47). Furthermore, employing (7.36b) and using that $\text {Im} \,\eta < \text {Im} \,\eta _0$ so that the contribution from $-\infty +ia$ vanishes, we obtain

$$\begin{aligned} \int _{\mathcal {C}_{\mathcal {Q}}^-} dx\; e^{\mathcal {A}_\mathcal {Q}^-(x,\eta )} = \int _{-\infty +ia}^{ia} dx\; e^{\mathcal {A}_\mathcal {Q}^-(x,\eta )} = e^{i\pi \left( \sigma _s^2+\frac{Q^2-2}{12}\right) } \frac{e^{-2\pi a(\eta -\eta _0)}}{2i\pi (\eta -\eta _0)}. \nonumber \\\end{aligned}$$

(7.48)

The right-hand side of (7.48) has a simple pole at $\eta =\eta _0$. Therefore, collecting the above conclusions,

$$\begin{aligned}{} & {} \lim \limits _{\eta \rightarrow \eta _0} \mathcal {Q}(b,\sigma _s,\eta ) = \lim \limits _{\eta \rightarrow \eta _0} P_\mathcal {Q}(\sigma _s,\eta ) \int _{\mathcal {C}_{\mathcal {Q}}^-} dx\; e^{\mathcal {A}_\mathcal {Q}^-(x,\eta )} \nonumber \\{} & {} \quad = \lim \limits _{\eta \rightarrow \eta _0} P_\mathcal {Q}(\sigma _s,\eta ) e^{i\pi \left( \sigma _s^2+\frac{Q^2-2}{12}\right) } \frac{e^{-2\pi a(\eta -\eta _0)}}{2i\pi (\eta -\eta _0)}, \end{aligned}$$

(7.49)

where the limits are taken with $\text {Im} \,\eta < \text {Im} \,\eta _0$. Utilizing (7.7) and the identity $s_b(z)=s_b(-z)^{-1}$, we obtain

$$\begin{aligned} \lim \limits _{\eta \rightarrow \eta _0} \mathcal {Q}(b,\sigma _s,\eta ) = \frac{1}{2i\pi } \lim \limits _{\eta \rightarrow \eta _0} \left( \frac{1}{s_b(-\eta )(\eta -\eta _0)}\right) . \end{aligned}$$

(7.50)

Setting $z=-\eta$, recalling that $\eta _0=iQ/2$, and using (2.4), we find

$$\begin{aligned} \lim \limits _{\eta \rightarrow \eta _0} \mathcal {Q}(b,\sigma _s,\eta ) = -\frac{1}{2i\pi } \lim \limits _{z\rightarrow -\tfrac{iQ}{2}} \left( \frac{1}{s_b(z)(z+\frac{iQ}{2})}\right) = 1, \end{aligned}$$

(7.51)

which proves (7.35) for $n=0$.

To show (7.35) also for $n \ge 1$, we rewrite the difference equation (7.31a) as follows:

$$\begin{aligned}{} & {} \mathcal {Q}(b,\sigma _s,\eta +ib) = - \left( 1 + e^{2\pi b(\eta -\frac{ib}{2})} \right) \mathcal {Q}(b,\sigma _s,\eta -ib)\nonumber \\{} & {} \quad +2{\text {cosh}}{(2\pi b \sigma _s)} \mathcal {Q}(b,\sigma _s,\eta ). \end{aligned}$$

(7.52)

Note that $1 + e^{2\pi b(\eta -\frac{ib}{2})}$ vanishes for $\eta =\eta _0$. Moreover, the function $\mathcal {Q}(b,\sigma _s,\eta -ib)$ is analytic at $\eta =\eta _0$. Indeed, as $\eta$ approaches $\eta _0$, the contour $\mathcal {C}_{\mathcal {Q}}$ remains above the two decreasing sequences of poles, and, in view of (7.36) (see also (7.43) and (7.41)), the integrand $I_\mathcal {Q}$ retains its exponential decay provided that the right tail of the contour is deformed downwards. Therefore, evaluating (7.52) at $\eta =\eta _0$ and using (7.51), we obtain $\mathcal {Q}(b,\sigma _s,\eta _1) = 2{\text {cosh}}{(2\pi b \sigma _s)}$. Evaluating the recurrence relation (B.41) satisfied by the continuous q-Hermite polynomials at $n=0$ and $z=e^{2\pi b \sigma _s}$, we find

$$\begin{aligned} Q_1\left( e^{2\pi b \sigma _s};q \right) = 2{\text {cosh}}{(2\pi b \sigma _s)} = \mathcal {Q}(b,\sigma _s,\eta _1), \end{aligned}$$

(7.53)

which proves (7.35) for $n=1$. More generally, suppose that the function $\mathcal {Q}(b,\sigma _s,\eta _n)$ exists for all $n\le N$ and coincides with the polynomials $Q_n\left( e^{2\pi b \sigma _s};q \right)$. Evaluating (7.52) at $n=N+1$, we obtain

$$\begin{aligned}{} & {} \mathcal {Q}(b,\sigma _s,\eta _{N+1}) = - \left( 1 -q^{N+1} \right) Q_N\left( e^{2\pi b \sigma _s};q \right) \nonumber \\{} & {} \quad +2{\text {cosh}}{(2\pi b \sigma _s)} Q_{N-1}\left( e^{2\pi b \sigma _s};q \right) , \end{aligned}$$

(7.54)

and hence the recurrence relation (B.41) implies that

$$\begin{aligned} \mathcal {Q}(b,\sigma _s,\eta _{N+1}) = Q_{N+1}\left( e^{2\pi b \sigma _s};q \right) . \end{aligned}$$

(7.55)

By induction, we conclude that $\mathcal {Q}(b,\sigma _s,\eta _n)$ exists for all $n\ge 0$ and coincides with $Q_n\left( e^{2\pi b \sigma _s};q \right)$. This completes the proof of the theorem. $\square$

8 The Function $\mathcal {L}$

In this section, we define the function $\mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu )$ which generalizes the big q-Laguerre polynomials. It is defined as a confluent limit of $\mathcal {H}$ and lies at the third level of the non-polynomial scheme. We show that $\mathcal {L}$ is a joint eigenfunction of four difference operators. Finally, we show that $\mathcal {L}$ reduces to the big q-Laguerre polynomials, which lie at the third level of the q-Askey scheme, when $\lambda$ is suitably discretized.

8.1 Definition and Integral Representation

Let $\theta , \Lambda , \lambda , \mu$ be defined as follows:

$$\begin{aligned} \theta _0 = \frac{\theta +\Lambda }{2}, \qquad \theta _*= \frac{\theta -\Lambda }{2}, \qquad \sigma _s = \lambda + \frac{\Lambda }{2}, \qquad \nu = \mu -\frac{\Lambda }{4}. \end{aligned}$$

(8.1)

Define the open set $\Omega _{\mathcal {L}} \subset \mathbb {C}^2$ by

$$\begin{aligned} \Omega _{\mathcal {L}}:= \; (\mathbb C \backslash \Delta _\lambda ) \times (\{\text {Im} \,\mu >-Q/2\} \backslash \Delta _\mu ), \end{aligned}$$

(8.2)

where the discrete subsets $\Delta _\lambda$ and $\Delta _\mu$ are given by

$$\begin{aligned} \Delta _\lambda :=&\; \{\tfrac{iQ}{2} \pm \theta _t + \tfrac{\theta }{2}+ibm+ilb^{-1} \}_{m,l=0}^\infty \cup \{\tfrac{iQ}{2} - \tfrac{\theta }{2}+ibm+ilb^{-1} \}_{m,l=0}^\infty \\&\cup \{-\tfrac{iQ}{2}+\theta _t + \tfrac{\theta }{2}-ibm-ilb^{-1} \}_{m,l=0}^\infty , \\ \Delta _\mu :=&\; \{\tfrac{iQ}{2} \pm \theta _t + \tfrac{\theta }{4}+ibm+ilb^{-1} \}_{m,l=0}^\infty \cup \{\tfrac{iQ}{2} - \tfrac{3\theta }{4}+ibm+ilb^{-1} \}_{m,l=0}^\infty \\&\cup \{-\tfrac{iQ}{2}+\theta _t + \tfrac{\theta }{4}-ibm-ilb^{-1} \}_{m,l=0}^\infty . \end{aligned}$$

Definition 8.1

Let $\mathcal {H}$ be defined by (4.2). The function $\mathcal {L}$ is defined for $(\lambda ,\mu ) \in \Omega _{\mathcal {L}}$ by

$$\begin{aligned} \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ) = \lim \limits _{\Lambda \rightarrow -\infty } \mathcal {H}\left( b, \tfrac{\theta +\Lambda }{2}, \theta _t, \tfrac{\theta -\Lambda }{2}, \lambda + \tfrac{\Lambda }{2}, \mu -\tfrac{\Lambda }{4}\right) , \end{aligned}$$

(8.3)

and is extended meromorphically to $(\lambda ,\mu ) \in \mathbb {C}^2$.

The next theorem shows that $\mathcal {L}$ is a well-defined meromorphic function of $(\lambda ,\mu ) \in \mathbb {C}^2$.

Theorem 8.2

Suppose that Assumption 1.1 is satisfied. The limit in (8.3) exists uniformly for $(\lambda ,\mu )$ in compact subsets of $\Omega _{\mathcal {L}}$. Moreover, $\mathcal L$ is an analytic function of $(\lambda ,\mu ) \in (\mathbb C \backslash \Delta _\lambda ) \times (\mathbb C \backslash \Delta _\mu )$ and admits the following integral representation:

$$\begin{aligned} \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ) = P_\mathcal {L}(\lambda ,\mu ) \displaystyle \int _{\mathcal {C}_{\mathcal {L}}} dx \; I_\mathcal {L}(x,\lambda ,\mu ) \qquad \text {for }(\lambda ,\mu ) \in (\mathbb C \backslash \Delta _\lambda ) \times (\mathbb C \backslash \Delta _\mu ), \nonumber \\ \end{aligned}$$

(8.4)

where

$$\begin{aligned} P_\mathcal {L}(\lambda ,\mu )&= s_b\left( 2 \theta _t+\tfrac{i Q}{2}\right) s_b\left( \theta +\theta _t+\tfrac{i Q}{2}\right) s_b\left( \lambda -\tfrac{\theta }{2}-\theta _t \right) s_b\left( \mu -\tfrac{\theta }{4}-\theta _t \right) , \end{aligned}$$

(8.5)

$$\begin{aligned} I_\mathcal {L}(x,\lambda ,\mu )&= e^{\frac{i\pi x^2}{2}} e^{i\pi x (\frac{\theta }{4}+\theta _t+\lambda +\mu -\frac{iQ}{2})} \nonumber \\&\quad \times \frac{s_b\left( x+\frac{\theta }{2}+\theta _t-\lambda \right) s_b\left( x+\frac{\theta }{4}+\theta _t-\mu \right) }{s_b\left( x+\frac{i Q}{2}\right) s_b\left( x+2 \theta _t+\frac{i Q}{2}\right) s_b\left( x+\theta +\theta _t+\frac{i Q}{2}\right) }, \end{aligned}$$

(8.6)

and the contour $\mathcal {C}_{\mathcal {L}}$ is any curve from $-\infty$ to $+\infty$ which separates the three increasing from the two decreasing sequences of poles, with the requirement that its right tail satisfies

$$\begin{aligned} \text {Im} \,x + \tfrac{Q}{2} + \text {Im} \,\lambda + \text {Im} \,\mu > \delta \qquad \text {for all }x \in \mathcal {C}_{\mathcal {L}}\text { with}\,\, \text {Re} \,x\text { sufficiently large}, \end{aligned}$$

(8.7)

for some $\delta >0$. In particular, $\mathcal L$ is a meromorphic function of $(\lambda ,\mu ) \in \mathbb C^2$. If $(\lambda ,\mu ) \in \mathbb R^2$, the contour $\mathcal {C}_{\mathcal {L}}$ can be any curve from $-\infty$ to $+\infty$ lying within the strip $\text {Im} \,x \in (-Q/2 +\delta ,0)$.

Furthermore, the function $\mathcal {L}$ obeys the symmetry

$$\begin{aligned} \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ) = \mathcal {L}(b^{-1},\theta _t,\theta ,\lambda ,\mu ). \end{aligned}$$

(8.8)

8.2 Difference Equations

The next two propositions, whose proofs are omitted because they are similar to those presented in Sect. 4.2, show that the two pairs of difference equations (4.26) and (4.33) satisfied by the function $\mathcal {H}$ survive in the confluent limit (8.3), implying that $\mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu )$ is a joint eigenfunction of four difference operators, two acting on $\lambda$ and the other two on $\mu$. The four difference equations hold as equalities between meromorphic functions of $(\lambda , \mu ) \in \mathbb C^2$.

8.2.1 First Pair of Difference Equations

Define a difference operator $D_\mathcal {L}(b,\lambda )$ such that

$$\begin{aligned} D_\mathcal {L}(b,\lambda ) = d^+_\mathcal {L}(b,\lambda ) e^{ib \partial _\lambda } + d^-_\mathcal {L}(b,\lambda ) e^{-ib \partial _\lambda } + d^0_\mathcal {L}(b,\lambda ), \end{aligned}$$

(8.9)

where

$$\begin{aligned}&d^+_\mathcal {L}(b,\lambda ) = -4 e^{-\pi b \left( \frac{\theta }{2}+\theta _t-2 \lambda \right) } \cosh \left( \pi b \left( \tfrac{i b}{2}+\tfrac{\theta }{2}+\lambda \right) \right) \cosh \left( \pi b \left( \tfrac{i b}{2}-\tfrac{\theta }{2}+\theta _t+\lambda \right) \right) , \end{aligned}$$

(8.10)

$$\begin{aligned}&d^-_\mathcal {L}(b,\lambda ) = -2 e^{\pi b \left( \theta _t-\frac{ib}{2}+3 \lambda \right) } \cosh \left( \pi b \left( \tfrac{i b}{2}+\tfrac{\theta }{2}+\theta _t-\lambda \right) \right) , \end{aligned}$$

(8.11)

and

$$\begin{aligned} d^0_\mathcal {L}(\lambda ) = e^{-\pi b \left( \frac{\theta }{2}+2 \theta _t+i Q\right) } - d^+_\mathcal {L}(b,\lambda ) - d^-_\mathcal {L}(b,\lambda ). \end{aligned}$$

(8.12)

Proposition 8.3

For $(\lambda ,\mu ) \in \mathbb C^2$, the function $\mathcal {L}$ satisfies the other pair of difference equations:

$$\begin{aligned}&D_\mathcal {L}(b,\lambda ) \; \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ) = e^{-2\pi b \mu } \; \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ), \end{aligned}$$

(8.13a)

$$\begin{aligned}&D_\mathcal {L}(b^{-1},\lambda ) \; \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ) = e^{-2\pi b^{-1} \mu } \; \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ). \end{aligned}$$

(8.13b)

8.2.2 Second Pair of Difference Equations

Define a difference operator $\tilde{D}_\mathcal {L}(b,\mu )$ such that

$$\begin{aligned} \tilde{D}_\mathcal {L}(b,\mu ) = \tilde{d}^+_\mathcal {L}(b,\mu ) e^{ib \partial _\mu } + \tilde{d}^-_\mathcal {L}(b,\mu ) e^{-ib \partial _\mu } + \tilde{d}^0_\mathcal {L}(b,\mu ), \end{aligned}$$

(8.14)

where

$$\begin{aligned}&\tilde{d}^+_\mathcal {L}(b,\mu ) = -4 e^{-\pi b \left( \frac{\theta }{2}+\theta _t-2 \mu \right) } \cosh \left( \pi b \left( \tfrac{i b}{2}+\tfrac{3 \theta }{4}+\mu \right) \right) \cosh \left( \pi b \left( \tfrac{i b}{2}-\tfrac{\theta }{4}+\theta _t+\mu \right) \right) , \end{aligned}$$

(8.15)

$$\begin{aligned}&\tilde{d}^-_\mathcal {L}(b,\mu ) = -2 e^{\pi b \left( -\frac{ib}{2}+\frac{\theta }{4}+\theta _t+3 \mu \right) } \cosh \left( \pi b \left( \tfrac{i b}{2}+\tfrac{\theta }{4}+\theta _t-\mu \right) \right) , \end{aligned}$$

(8.16)

and

$$\begin{aligned} \tilde{d}^0_\mathcal {L}(b,\mu ) = e^{-\pi b (\theta +2 \theta _t+i Q)} - \tilde{d}^+_\mathcal {L}(b,\mu ) - \tilde{d}^-_\mathcal {L}(b,\mu ). \end{aligned}$$

(8.17)

Proposition 8.4

For $(\lambda ,\mu ) \in \mathbb C^2$, the function $\mathcal {L}$ satisfies the pair of difference equations

$$\begin{aligned}&\tilde{D}_\mathcal {L}(b,\mu ) \; \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ) = e^{-2\pi b \lambda } \; \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ), \end{aligned}$$

(8.18a)

$$\begin{aligned}&\tilde{D}_\mathcal {L}(b^{-1},\mu ) \; \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ) = e^{-2\pi b^{-1} \lambda } \; \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ). \end{aligned}$$

(8.18b)

8.3 Polynomial Limit

Our next theorem, whose proof is similar to that of Theorem 4.7, shows that the function $\mathcal {L}$ reduces to the big q-Laguerre polynomials $L_n$ when $\lambda$ is suitably discretized.

The following assumption ensures that all the poles of the integrand in (8.4) are distinct and simple.

Assumption 8.5

Assume that $b>0$ is such that $b^2$ is irrational and that

$$\begin{aligned} \theta _t \ne 0, \qquad \theta \pm \theta _t \ne 0, \qquad \text {Re} \,\big (\tfrac{\theta }{4} -\lambda + \mu \big ) \ne 0. \end{aligned}$$

(8.19)

Theorem 8.6

(From $\mathcal {L}$ to the big q-Laguerre polynomials) Let $\mu \in \{\text {Im} \,\mu >-Q/2\} \backslash \Delta _\mu$ and suppose that Assumptions 1.1 and 8.5 are satisfied. Define $\{\lambda _n\}_{n=0}^\infty \subset \mathbb {C}$ by

$$\begin{aligned} \lambda _n = \frac{\theta }{2}+\theta _t+\tfrac{i Q}{2}+ibn. \end{aligned}$$

(8.20)

Under the parameter correspondence

$$\begin{aligned} \alpha _{L} = e^{4\pi b \theta _t}, \qquad \beta _{L} = e^{2\pi b(\theta +\theta _t)}, \qquad x_{L}= e^{\pi b(iQ+\frac{\theta }{2}+2\theta _t)} e^{-2\pi b \mu }, \qquad q=e^{2i\pi b^2},\nonumber \\ \end{aligned}$$

(8.21)

the function $\mathcal {L}$ satisfies, for each $n\ge 0$,

$$\begin{aligned} \lim \limits _{\lambda \rightarrow \lambda _n} \mathcal {L}(b,\theta _t,\theta ,\lambda ,\mu ) = L_n(x_{L};\alpha _{L},\beta _{L},q), \end{aligned}$$

(8.22)

where $L_n$ are the big q-Laguerre polynomials defined in (B.29).

9 The Function $\mathcal {W}$

In this section, we define the function $\mathcal {W}(b,\theta _t,\kappa ,\omega )$ which generalizes the little q-Laguerre polynomials. It is defined as a confluent limit of $\mathcal {L}$ and lies at the fourth level of the non-polynomial scheme. We show that $\mathcal {W}$ is a joint eigenfunction of four difference operators and that it reduces to the little q-Laguerre polynomials, which lie at the fourth level of the q-Askey scheme, when $\kappa$ is suitably discretized.

9.1 Definition and Integral Representation

Introduce two new parameters $\kappa$ and $\omega$ by

$$\begin{aligned} \lambda = \frac{\theta }{2}+\kappa , \qquad \mu = -\frac{3\theta }{4}+\omega . \end{aligned}$$

(9.1)

Definition 9.1

The function $\mathcal {W}(b,\theta _t,\kappa ,\omega )$ is defined by

$$\begin{aligned} \mathcal {W}(b,\theta _t,\kappa ,\omega ) = \lim \limits _{\theta \rightarrow +\infty } \mathcal {L}(b,\theta _t,\theta ,\tfrac{\theta }{2}+\kappa , -\tfrac{3\theta }{4}+\omega ), \end{aligned}$$

(9.2)

where $\mathcal {L}$ is defined by (8.4).

The next theorem, whose proof is similar to that of Theorem 4.2 and will be omitted, shows that, for each choice of $(b,\theta _t) \in (0,\infty ) \times \mathbb R$, $\mathcal W$ is a well-defined and analytic function of $(\kappa ,\omega ) \in (\mathbb {C} \backslash \Delta _\kappa ) \times \{\text {Im} \,\omega < Q/2 \}$, where $\Delta _\kappa \subset \mathbb C$ is a discrete set of points at which $\mathcal W$ may have poles. In particular, $\mathcal W$ is a meromorphic function of $\kappa \in \mathbb C$ and of $\omega$ for $\text {Im} \,\omega < Q/2$. The theorem also provides an integral representation for $\mathcal W$ for $(\kappa ,\omega ) \in (\mathbb {C} \backslash \Delta _\kappa ) \times \{\text {Im} \,\omega < Q/2 \}$. In fact, even if the requirement $\text {Im} \,\omega < Q/2$ is needed to ensure that the integral representation converges, it will be shown later, with the help of the difference equations, that $\mathcal W$ extends to a meromorphic function of $(\kappa ,\omega ) \in \mathbb C^2$.

Theorem 9.2

Suppose that Assumption 1.1 holds. The limit (9.2) exists uniformly for $(\kappa ,\omega )$ in compact subsets of

$$\begin{aligned} \Omega _{\mathcal {W}}:= \; (\mathbb C \backslash \Delta _\kappa ) \times \{\text {Im} \,\omega < Q/2 \}, \end{aligned}$$

(9.3)

where

$$\begin{aligned} \Delta _\kappa := \{\tfrac{iQ}{2} \pm \theta _t+imb+ilb^{-1} \}_{m,l=0}^\infty \cup \{-\tfrac{iQ}{2}+\theta _t-imb-ilb^{-1} \}_{m,l=0}^\infty . \nonumber \\\end{aligned}$$

(9.4)

Moreover, $\mathcal {W}$ is an analytic function of $(\kappa ,\omega ) \in \Omega _{\mathcal {W}}$ and admits the following integral representation:

$$\begin{aligned} \mathcal {W}(b,\theta _t,\kappa ,\omega ) = P_\mathcal {W}(\kappa ,\omega ) \displaystyle \int _{\mathcal {C}_{\mathcal {W}}} dx \; I_\mathcal {W}(x,\kappa ,\omega ) \qquad \text {for }(\kappa ,\omega ) \in \Omega _{\mathcal {W}}, \end{aligned}$$

(9.5)

where

$$\begin{aligned}&P_\mathcal {W}(\kappa ,\omega ) = s_b(\tfrac{iQ}{2}+2\theta _t) s_b(\kappa -\theta _t), \end{aligned}$$

(9.6)

$$\begin{aligned}&I_\mathcal {W}(x,\kappa ,\omega ) = e^{\frac{i \pi x^2}{2}} e^{i \pi x (\theta _t+\kappa +2 \omega )} \frac{s_b(x+\theta _t-\kappa )}{s_b\left( x+\frac{i Q}{2}\right) s_b\left( x+2 \theta _t +\frac{i Q}{2}\right) }, \end{aligned}$$

(9.7)

and the contour $\mathcal {C}_{\mathcal {W}}$ is any curve from $-\infty$ to $+\infty$ which separates the decreasing sequence of poles from the two increasing ones, with the requirement that its right tail satisfies

$$\begin{aligned} \text {Im} \,x + \tfrac{Q}{2} + \text {Im} \,\kappa + \text {Im} \,\omega > \delta \qquad \text {for all }x \in \mathcal {C}_{\mathcal {W}}\text { with }\text {Re} \,x\text { sufficiently large}, \nonumber \\\end{aligned}$$

(9.8)

for some $\delta >0$. If $(\kappa ,\omega )\in \mathbb R^2$, then $\mathcal {C}_{\mathcal {W}}$ can be any curve from $-\infty$ to $+\infty$ lying within the strip $\text {Im} \,x \in (-Q/2+\delta ,0)$.

Furthermore, thanks to the symmetry (2.7), $\mathcal W$ satisfies

$$\begin{aligned} \mathcal {W}(b^{-1},\theta _t,\kappa ,\omega ) = \mathcal {W}(b,\theta _t,\kappa ,\omega ). \end{aligned}$$

(9.9)

9.2 Difference Equations

By taking the confluent limit (9.2) of the difference equations (8.3) and (8.4) satisfied by $\mathcal {L}$, we obtain the next two propositions which show that $\mathcal {W}$ is a joint eigenfunction of four difference operators, two acting on $\kappa$ and the other two on $\omega$.

The difference equations will first be derived as equalities between meromorphic functions of $\kappa \in \mathbb C$ and $\omega$ with $\text {Im} \,\omega < Q/2$. We will then use the difference equations in $\omega$ to show that (i) the limit in (9.2) exists for all $\omega$ in the complex plane away from a discrete subset, (ii) $\mathcal {W}$ is in fact a meromorphic function of $(\kappa ,\omega )$ in all of $\mathbb C^2$, and (iii) the four difference equations hold as equalities between meromorphic functions on $\mathbb C^2$, see Proposition 9.5.

9.2.1 First Pair of Difference Equations

Define the difference operator $D_{\mathcal {W}}(b,\kappa )$ such that

$$\begin{aligned} D_{\mathcal {W}}(b,\kappa ) = d^+_{\mathcal {W}}(b,\kappa ) e^{ib \partial _\kappa } + d^-_{\mathcal {W}}(b,\kappa ) e^{-ib \partial _\kappa } + d^0_{\mathcal {W}}(b,\kappa ), \end{aligned}$$

(9.10)

where

$$\begin{aligned}&d^\pm _{\mathcal {W}}(b,\kappa ) = -2 e^{3\pi b \kappa } e^{\pm \pi b(\frac{ib}{2}-\theta _t)} \cosh \left( \pi b \left( \tfrac{i b}{2}+\theta _t\pm \kappa \right) \right) , \end{aligned}$$

(9.11)

$$\begin{aligned}&d^0_{\mathcal {W}}(b,\kappa ) = - d^+_{\mathcal {W}}(b,\kappa ) - d^-_{\mathcal {W}}(b,\kappa ). \end{aligned}$$

(9.12)

Proposition 9.3

For $\kappa \in \mathbb C$ and $\text {Im} \,\omega <Q/2$, the function $\mathcal {W}$ satisfies the following pair of difference equations:

$$\begin{aligned}&D_{\mathcal {W}}(b,\kappa )\; \mathcal {W}(b,\theta _t,\kappa ,\omega ) = e^{-2\pi b \omega }\; \mathcal {W}(b,\theta _t,\kappa ,\omega ), \end{aligned}$$

(9.13a)

$$\begin{aligned}&D_{\mathcal {W}}(b^{-1},\kappa )\; \mathcal {W}(b,\theta _t,\kappa ,\omega ) = e^{-2\pi b^{-1} \omega }\; \mathcal {W}(b,\theta _t,\kappa ,\omega ). \end{aligned}$$

(9.13b)

9.2.2 Second Pair of Difference Equations

Introduce the difference operator $\tilde{D}_{\mathcal {W}}(b,\omega )$ by

$$\begin{aligned} \tilde{D}_{\mathcal {W}}(b,\omega ) = \tilde{d}_{\mathcal {W}}^+(b,\omega ) e^{ib \partial _\omega } + \tilde{d}_{\mathcal {W}}^-(b,\omega ) e^{-ib \partial _\omega } + \tilde{d}_{\mathcal {W}}^0(b,\omega ), \end{aligned}$$

(9.14)

where

$$\begin{aligned}&\tilde{d}_{\mathcal {W}}^+(b,\omega ) = -2 e^{\pi b \left( \omega -\frac{i b}{2}-2 \theta _t \right) } \cosh \left( \pi b \left( \omega +\tfrac{i b}{2}\right) \right) , \end{aligned}$$

(9.15)

$$\begin{aligned}&\tilde{d}_{\mathcal {W}}^-(b,\omega ) = -e^{2 \pi b (\theta _t+\omega )}, \end{aligned}$$

(9.16)

$$\begin{aligned}&\tilde{d}_{\mathcal {W}}^0(b,\omega ) = 2 e^{2 \pi b \omega } \cosh (2 \pi b \theta _t). \end{aligned}$$

(9.17)

Proposition 9.4

For $\kappa \in \mathbb C$ and $\text {Im} \,(\omega +ib^{\pm 1}) <Q/2$, the function $\mathcal {W}$ satisfies the following pair of difference equations:

$$\begin{aligned}&\tilde{D}_{\mathcal {W}}(b,\omega )\;\mathcal {W}(b,\theta _t,\kappa ,\omega ) = e^{-2\pi b \kappa }\; \mathcal {W}(b,\theta _t,\kappa ,\omega ), \end{aligned}$$

(9.18a)

$$\begin{aligned}&\tilde{D}_{\mathcal {W}}(b^{-1},\omega )\;\mathcal {W}(b,\theta _t,\kappa ,\omega ) = e^{-2\pi b^{-1} \kappa }\;\mathcal {W}(b,\theta _t,\kappa ,\omega ). \end{aligned}$$

(9.18b)

The next proposition is stated without proof since it is similar to that of Proposition 4.5.

Proposition 9.5

Let $(b,\theta _t) \in (0,\infty ) \times \mathbb R$ and $\kappa \in \mathbb C \backslash \Delta _\kappa$. There is a discrete subset $\Delta \subset \mathbb C$ such that the limit in (9.2) exists for all $\omega \in \mathbb C \backslash \Delta$. Moreover, the function $\mathcal W$ defined by (9.2) is a meromorphic function of $(\kappa ,\omega ) \in \mathbb C^2$ and the four difference equations (9.3) and (9.4) hold as equalities between meromorphic functions of $(\kappa ,\omega ) \in \mathbb C^2$.

9.3 Polynomial Limit

Our next theorem shows that $\mathcal {W}$ reduces to the little q-Laguerre polynomials when $\kappa$ is suitably discretized. We omit the proof which is similar to that of Theorem 4.7.

Assumption 9.6

Assume that $b>0$ is such that $b^2$ is irrational and

$$\begin{aligned} \theta _t \ne 0. \end{aligned}$$

(9.19)

Assumption 9.6 implies that all the poles of the integrand $I_\mathcal {W}$ are distinct and simple.

Theorem 9.7

(From $\mathcal {W}$ to the little q-Laguerre polynomials) Let $\omega \in \mathbb {C}$ be such that $\text {Im} \,\omega < Q/2$ and suppose that Assumptions 1.1 and 9.6 are satisfied. Define $\{\kappa _n\}_{n=0}^\infty \subset \mathbb {C}$ by

$$\begin{aligned} \kappa _n = \theta _t + \tfrac{i Q}{2} + ibn. \end{aligned}$$

(9.20)

Under the parameter correspondence

$$\begin{aligned} \alpha _{W} = e^{4\pi b \theta _t}, \qquad x_{W}=e^{-2 \pi b \left( \frac{i Q}{2}+\omega \right) },\qquad q=e^{2i\pi b^2}, \end{aligned}$$

(9.21)

the function $\mathcal {W}$ satisfies, for each $n\ge 0$,

$$\begin{aligned} \lim \limits _{\kappa \rightarrow \kappa _n} \mathcal {W}(b,\theta _t,\kappa ,\omega ) = W_n(x_{W};\alpha _{W},q), \end{aligned}$$

(9.22)

where $W_n$ are the little q-Laguerre polynomials defined in (B.33).

10 The Function $\mathcal {M}$

In this section, we define the function $\mathcal {M}(b,\zeta ,\omega )$ which generalizes the little q-Laguerre polynomials (B.33) evaluated at $\alpha =0$. It lies at the fifth and lowest level of the non-polynomial scheme and is defined as a limit of the function $\mathcal {W}$. We show that $\mathcal M$ is a joint eigenfunction of four difference operators, two acting on $\zeta$ and the other two on $\omega$. Finally, we show that $\mathcal M$ reduces to the little q-Laguerre polynomials evaluated at $\alpha =0$ when $\zeta$ is suitably discretized.

10.1 Definition and Integral Representation

Introduce a new parameter $\zeta$ such that $\kappa = \theta _t+\zeta$.

Definition 10.1

The function $\mathcal {M}(b,\zeta ,\omega )$ is defined by

$$\begin{aligned} \mathcal {M}(b,\zeta ,\omega ) = \lim \limits _{\theta _t \rightarrow -\infty } \mathcal {W}(b,\theta _t,\theta _t + \zeta ,\omega ), \end{aligned}$$

(10.1)

where the function $\mathcal {W}$ is defined in (9.2).

The next theorem shows that, for each choice of $b \in (0,\infty )$, $\mathcal M$ is a well-defined and analytic function of

$$\begin{aligned} (\zeta ,\omega ) \in \{\text {Im} \,\omega <Q/2,\; \text {Im} \,(\zeta +\omega ) >0 \} \setminus (\Delta _\zeta \times \mathbb C), \end{aligned}$$

(10.2)

where $\Delta _\zeta \subset \mathbb C$ is a discrete set of points at which $\mathcal M$ may have poles. More precisely, $\Delta _\zeta$ is defined by

$$\begin{aligned} \Delta _{\zeta }:= \{\tfrac{iQ}{2}+ibm+ilb^{-1}\}_{m,l=0}^\infty \cup \{-\tfrac{iQ}{2}-imb-ilb^{-1} \}_{m,l=0}^\infty . \end{aligned}$$

(10.3)

The theorem also provides an integral representation for $\mathcal M$ for $(\zeta ,\omega )$ satisfying (10.2). The restrictions in (10.2) are needed to ensure convergence of the integral in the integral representation. Nevertheless, it will be shown later, with the help of the difference equations satisfied by $\mathcal M$, that $\mathcal M$ extends to a meromorphic function of $(\zeta ,\omega ) \in \mathbb C^2$.

Theorem 10.2

Suppose that Assumption 1.1 holds. Let $\Delta _\zeta \subset \mathbb C$ be the discrete subset defined in (10.3). Then the limit (10.1) exists uniformly for $(\zeta ,\omega )$ in compact subsets of

$$\begin{aligned} \Omega _\mathcal {M}:= \{(\zeta , \omega ) \in \mathbb C^2 \,|\, \text {Im} \,\omega <Q/2,\; \text {Im} \,(\zeta +\omega ) >0 \} \setminus (\Delta _\zeta \times \mathbb C). \end{aligned}$$

(10.4)

Moreover, $\mathcal M$ is an analytic function of $(\zeta ,\omega ) \in \Omega _\mathcal {M}$ and admits the following integral representation:

$$\begin{aligned} \mathcal {M}(b,\zeta ,\omega ) = P_\mathcal {M}(\zeta ,\omega ) \displaystyle \int _{\mathcal {C}_{\mathcal {M}}} dx \; I_\mathcal {M}(x,\zeta ,\omega ) \qquad \text {for }(\zeta ,\omega ) \in \Omega _{\mathcal {M}}, \end{aligned}$$

(10.5)

where

$$\begin{aligned}&P_\mathcal {M}(\zeta ,\omega ) = s_b(\zeta ), \end{aligned}$$

(10.6)

$$\begin{aligned}&I_\mathcal {M}(x,\zeta ,\omega ) = e^{i \pi x \left( \zeta -\frac{i Q}{2}+2 \omega \right) } \frac{s_b(x-\zeta ) }{s_b\big (x+\frac{i Q}{2}\big )}, \end{aligned}$$

(10.7)

and the contour $\mathcal {C}_{\mathcal {M}}$ is any curve from $-\infty$ to $+\infty$ which separates the increasing from the decreasing sequence of poles. If $\text {Im} \,\omega \in (0,Q/2)$ and $\zeta \in \mathbb R$, then the contour $\mathcal {C}_{\mathcal {M}}$ can be any curve from $-\infty$ to $+\infty$ lying within the strip $\text {Im} \,x \in (-Q/2,0)$.

Thanks to the identity (2.7), $\mathcal {M}$ satisfies

$$\begin{aligned} \mathcal {M}(b^{-1},\zeta ,\omega ) = \mathcal {M}(b,\zeta ,\omega ). \end{aligned}$$

(10.8)

10.2 Difference Equations

The two pairs of difference equations (9.3) and (9.4) satisfied by the function $\mathcal {W}$ survive in the confluent limit (10.1). This implies that $\mathcal {M}$ is a joint eigenfunction of four difference operators, two acting on $\zeta$ and two acting on $\omega$.

We know from Theorem 10.2 that $\mathcal M$ is a well-defined holomorphic function of $\omega$ for $\text {Im} \,\omega <Q/2$ and is meromorphic in $\zeta$ for $\text {Im} \,(\zeta +\omega ) >0$. The difference equations will first be derived as equalities between meromorphic functions defined on this limited domain and then extended to equalities between meromorphic functions on $\mathbb C^2$, see Proposition 10.5.

10.2.1 First Pair of Difference Equations

Consider the difference operator $D_\mathcal {M}(b,\omega )$ defined by

$$\begin{aligned} D_\mathcal {M}(b,\zeta ) = e^{2\pi b \zeta } \left( 1 - e^{ib\partial _\zeta } \right) . \end{aligned}$$

(10.9)

Proposition 10.3

For $\text {Im} \,\omega <Q/2$ and $\text {Im} \,(\zeta +\omega )>0$, the function $\mathcal {M}$ satisfies the difference equations

$$\begin{aligned}&D_\mathcal {M}(b,\zeta )\;\mathcal {M}(b,\zeta ,\omega ) = e^{-2\pi b \omega }\; \mathcal {M}(b,\zeta ,\omega ), \end{aligned}$$

(10.10a)

$$\begin{aligned}&D_\mathcal {M}(b^{-1},\zeta )\;\mathcal {M}(b,\zeta ,\omega ) = e^{-2\pi b^{-1} \omega }\; \mathcal {M}(b,\zeta ,\omega ). \end{aligned}$$

(10.10b)

Proof

The proof consists of taking the confluent limit (10.1) of the difference equation (9.13a). It is straightforward to verify that

$$\begin{aligned} \lim \limits _{\theta _t \rightarrow -\infty } d_\mathcal {W}^+(b,\theta _t+\zeta ) = -e^{2\pi b \zeta }, \qquad \lim \limits _{\theta _t \rightarrow -\infty } d_\mathcal {W}^-(b,\theta _t+\zeta ) = 0, \end{aligned}$$

(10.11)

where $d_\mathcal {W}^\pm$ is defined in (9.11). From (9.12), this implies

$$\begin{aligned} \lim \limits _{\theta _t \rightarrow -\infty } d_\mathcal {W}^0(b,\theta _t+\zeta ) = e^{2\pi b \zeta }. \end{aligned}$$

(10.12)

Therefore we obtain

$$\begin{aligned} \lim \limits _{\theta _t \rightarrow -\infty } D_\mathcal {W}(b,\theta _t+\zeta ) = D_\mathcal {M}(b,\zeta ), \end{aligned}$$

(10.13)

where $D_\mathcal {M}$ is given in (10.9). By Theorem 10.2, the limit in (10.1) exists whenever $(\zeta ,\omega ) \in D_\mathcal {M}$. Thus, the difference equation (10.10a) follows after utilizing (10.13) and the definition (10.1) of $\mathcal M$. Finally, (10.10b) follows from (10.10a) and the symmetry (10.8) of $\mathcal {M}$. $\square$

10.2.2 Second Pair of Difference Equations

Define the dual difference operator $\tilde{D}_\mathcal {M}(b,\omega )$ by

$$\begin{aligned} \tilde{D}_\mathcal {M}(b,\omega ) = e^{2\pi b \omega } - 2 e^{\pi b (\omega -\frac{ib}{2})} {\text {cosh}}{(\pi b(\tfrac{ib}{2}+\omega ))} e^{ib \partial _\omega }. \end{aligned}$$

(10.14)

Proposition 10.4

For $\text {Im} \,(\omega +ib^{\pm 1}) < Q/2$ and $\text {Im} \,(\zeta +\omega )>0$, the function $\mathcal {M}$ satisfies the following pair of difference equations:

$$\begin{aligned}&\tilde{D}_\mathcal {M}(b,\omega )\;\mathcal {M}(b,\zeta ,\omega ) = e^{-2\pi b \zeta }\; \mathcal {M}(b,\zeta ,\omega ), \end{aligned}$$

(10.15a)

$$\begin{aligned}&\tilde{D}_\mathcal {M}(b^{-1},\omega )\;\mathcal {M}(b,\zeta ,\omega ) = e^{2\pi b^{-1} \zeta }\; \mathcal {M}(b,\zeta ,\omega ). \end{aligned}$$

(10.15b)

Proof

It is straightforward to show that the following limit holds:

$$\begin{aligned} e^{2\pi b \theta _t} \tilde{D}_\mathcal {W}(b,\omega ) = \tilde{D}_\mathcal {M}(b,\omega ), \end{aligned}$$

(10.16)

where $\tilde{D}_\mathcal {W}$ and $\tilde{D}_\mathcal {M}$ are defined by (9.14) and (10.14), respectively. By Theorem 10.2, the limit in (10.1) exists whenever $(\zeta ,\omega ) \in D_\mathcal {M}$. Thus, the difference equation (10.15a) follows after multiplying (9.18a) by $e^{2\pi b \theta _t}$ and utilizing (10.16) and the definition (10.1) of $\mathcal M$. Finally, (10.15b) follows from (10.15a) thanks to the symmetry (10.8) of $\mathcal M$. $\square$

Proposition 10.5

Let $b \in (0,\infty )$. Then there exist discrete subsets $\Delta ,\Delta ' \subset \mathbb C$ such that the limit in (10.1) exists for all $(\zeta ,\omega ) \in (\mathbb C {\setminus } \Delta ) \times (\mathbb C {\setminus } \Delta ')$. Moreover, the function $\mathcal M$ defined by (10.5) is a meromorphic function of $(\zeta ,\omega ) \in \mathbb C^2$ and the four difference equations (10.10) and (10.15) hold as equalities between meromorphic functions of $(\zeta ,\omega ) \in \mathbb C^2$.

Proof

The proof utilizes the difference equations in $\zeta$ and $\omega$ and is similar to the proof of Proposition 4.5. $\square$

10.3 Polynomial Limit

We now show that the function $\mathcal {M}$ reduces to the little q-Laguerre polynomials with $\alpha =0$ when $\zeta$ is suitably discretized.

Theorem 10.6

(From $\mathcal {M}$ to the little q-Laguerre polynomials with $\alpha =0$) Assume that $b>0$ is such that $b^2$ is irrational. Suppose $\omega \in \mathbb {C}$ satisfies $-Q/2+\delta< \text {Im} \,\omega < Q/2$ for some $\delta > 0$. Define $\{\zeta _n\}_{n=0}^\infty \subset \mathbb {C}$ by

$$\begin{aligned} \zeta _n = \frac{iQ}{2}+ibn. \end{aligned}$$

(10.17)

For each $n\ge 0$, the function $\mathcal M$ satisfies

$$\begin{aligned} \lim \limits _{\zeta \rightarrow \zeta _n} \mathcal {M}(b,\zeta ,\omega ) = W_n(x_W;0,q), \end{aligned}$$

(10.18)

where $W_n$ are the little q-Laguerre polynomials defined in (B.33) and where $x_W, q$ are given in (9.21).

Proof

We prove (10.18) by computing the limit $\zeta \rightarrow \zeta _n$ of the representation (10.5) for each $n\ge 0$. Let $m,l\ge 0$ be integers and define $x_{m,l} \in \mathbb {C}$ by

$$\begin{aligned} x_{m,l} = \zeta -\frac{iQ}{2}-imb-\frac{il}{b}. \end{aligned}$$

(10.19)

The function $s_b(x-\zeta )$ in (10.7) has a simple pole located at $x=x_{m,l}$ for any integers $m,l \ge 0$. In the limit $\zeta \rightarrow \zeta _n$, the pole $x_{n,0}$ collides with the pole of $s_b(x+\tfrac{iQ}{2})$ located at $x=0$, and the contour $\mathcal {C}_{\mathcal {M}}$ is squeezed between the colliding poles. Hence, before taking the limit $\zeta \rightarrow \zeta _n$, we deform $\mathcal {C}_{\mathcal {M}}$ into a contour $\mathcal {C}_{\mathcal {M}}'$ which passes below $x_{n,0}$, thus picking up residue contributions from all the poles $x=x_{m,l}$ which satisfy $\text {Im} \,x_{m,l} \ge \text {Im} \,x_{n,0}$, i.e., from all the poles $x_{m,l}$ such that $mb+\tfrac{l}{b} \le nb$. This leads to

$$\begin{aligned}&\mathcal {M}(b,\zeta ,\omega ) = -2i\pi P_\mathcal {M}(\zeta ,\omega ) \displaystyle \sum _{\begin{array}{c} m,l \ge 0 \\ mb+\tfrac{l}{b} \le nb \end{array} } \underset{x=x_{m,l}}{\text {Res}} \left( I_\mathcal {M}(x,\zeta ,\omega ) \right) \nonumber \\&\quad + P_\mathcal {M}(\zeta ,\omega ) \displaystyle \int _{\mathcal {C}_{\mathcal {M}}'} dx \; I_\mathcal {M}(x,\zeta ,\omega ). \end{aligned}$$

(10.20)

Utilizing the generalized difference equation (2.6) satisfied by the function $s_b$ and the residue formula (2.4), straightforward calculations show that

$$\begin{aligned}&-2i\pi \underset{x=x_{m,l}}{\text {Res}} \left( I_\mathcal {M}(x,\zeta ,\omega ) \right) \nonumber \\&\quad = e^{-i \pi \left( \frac{b^2 m^2}{2}+b m (Q+i (\zeta +2 \omega ))+\left( \zeta -\frac{i Q}{2}\right) \left( \frac{i Q}{2}-\zeta -2 \omega \right) \right) } e^{-i\pi b^{-2}(\frac{l^2}{2}+i b l (\zeta -i b m -i Q+2 \omega ))} \nonumber \\&\times \frac{1}{\big (e^{-\frac{2 i l \pi }{b^2}};e^{\frac{2 i \pi }{b^2}}\big ){}_l \left( e^{-2 i b^2 m \pi };e^{2 i b^2 \pi }\right) {}_m } \frac{1}{s_b\left( \zeta -\frac{i l}{b}-i b m \right) }. \end{aligned}$$

(10.21)

Because of the factor $s_b(\zeta -\frac{i l}{b}-i b m )^{-1}$, we deduce from (2.3) that the right-hand side of (10.21) has a simple pole at $\zeta =\zeta _n$ if the pair (m, l) satisfies $m \in [0,n]$ and $l=0$, but is regular at $\zeta =\zeta _n$ for all other choices of $m\ge 0$ and $l \ge 0$. On the other hand, the prefactor $P_\mathcal {M}$ in (10.6) has a simple zero at $\zeta =\zeta _n$. Therefore, in the limit $\zeta \rightarrow \zeta _n$ the second term on the right-hand side of (10.20) vanishes, and the first term is nonzero only if $m \in [0,n]$ and $l=0$. We conclude that

$$\begin{aligned}{} & {} \lim \limits _{\zeta \rightarrow \zeta _n} \mathcal {M}(b,\zeta ,\omega ) \nonumber \\{} & {} \quad = \mathcal {M}(b,\zeta _n,\omega ) = -2i\pi \lim \limits _{\zeta \rightarrow \zeta _n} P_\mathcal {M}(\zeta ,\omega ) \displaystyle \sum _{m=0}^n \underset{x=x_{m,0}}{\text {Res}} \left( I_\mathcal {M}(x,\zeta ,\omega ) \right) , \nonumber \\\end{aligned}$$

(10.22)

or, more explicitly,

$$\begin{aligned}{} & {} \mathcal {M}(b,\zeta _n,\omega ) \nonumber \\{} & {} \quad = \displaystyle \sum _{m=0}^n e^{-i \pi b^2\left( \frac{m^2}{2}-mn+n^2 \right) }e^{2\pi b\left( \omega (m-n)-\frac{imQ}{4}\right) } \frac{1}{\left( e^{-2 i b^2 m \pi };e^{2 i b^2 \pi }\right) {}_m} \frac{s_b(ibn+\frac{iQ}{2})}{s_b(ib(n-m)+\frac{iQ}{2})}.\nonumber \\ \end{aligned}$$

(10.23)

Using (2.6), we obtain

$$\begin{aligned} \mathcal {M}(b,\zeta _n,\omega ) = \displaystyle \sum _{m=0}^n e^{\pi b \left( 2(m-n) \omega -i b n^2-i m Q\right) } \frac{\left( q^{1+n-m};q\right) {}_m}{\left( q^{-m};q\right) {}_m}. \end{aligned}$$

(10.24)

We now apply the q-Pochhammer identity

$$\begin{aligned} (\alpha ;q)_n = (-\alpha )^n q^{\frac{n(n-1)}{2}} \left( \alpha ^{-1} q^{1-n};q \right) _n \end{aligned}$$

(10.25)

with $\alpha =q^{1+n-m}$ and with $\alpha =q^{-m}$ to find

$$\begin{aligned} \mathcal {M}(b,\zeta _n,\omega ) = \displaystyle \sum _{m=0}^n e^{\pi b (2 m-n) (2 \omega +i b n)} e^{i \pi m \left( b^2+2 i b \omega -1\right) } \frac{\big (q^{-n};q\big ){}_m}{\left( q;q\right) {}_m}. \end{aligned}$$

(10.26)

After replacing $m \rightarrow n-m$, in the sum and using the parameter correspondence (9.21), we obtain

$$\begin{aligned} \mathcal {M}(b,\zeta _n,\omega ) = \displaystyle \sum _{m=0}^n e^{i\pi (2m-n)} e^{-i\pi b^2(2m-n)(n+1)} \frac{\left( q^{-n};q\right) {}_{n-m}}{\left( q;q\right) {}_{n-m}} (q x_W)^m. \end{aligned}$$

(10.27)

Using the q-Pochhammer identity

$$\begin{aligned} \frac{(\alpha ;q)_{n-m}}{(\beta ;q)_{n-m}} = \frac{(\alpha ;q)_n}{(\beta ;q)_n} \frac{(\beta ^{-1} q^{1-n};q)_m}{(\alpha ^{-1} q^{1-n};q)_m} \left( \frac{\beta }{\alpha }\right) ^m, \qquad \alpha ,\beta \ne 0, \quad m=0,1,\dots ,n,\nonumber \\ \end{aligned}$$

(10.28)

with $\alpha =q^{-n}$ and $\beta =q$, we arrive at

$$\begin{aligned} \mathcal {M}(b,\zeta _n,\omega ) = \displaystyle e^{i \pi n \left( b^2 (n+1)-1\right) } \frac{\left( q^{-n};q\right) {}_n}{\left( q;q\right) {}_n} \sum _{m=0}^n \frac{\left( q^{-n};q\right) {}_m}{\left( q;q\right) {}_m} (q x_W)^m. \end{aligned}$$

(10.29)

Finally, utilizing the q-Pochhammer identity (10.25) with $\alpha =q^{-n}$, we find

$$\begin{aligned} e^{i \pi n \left( b^2 (n+1)-1\right) } \frac{\left( q^{-n};q\right) {}_n}{\left( q;q\right) {}_n} = 1. \end{aligned}$$

(10.30)

Therefore we conclude that

$$\begin{aligned} \mathcal {M}(b,\zeta _n,\omega ) = {}_2\phi _1\left( \left. \begin{aligned} \begin{matrix} q^{-n}, 0 \\ 0 \end{matrix} \end{aligned} \right| q;q x_W\right) = W_n(x_W;0,q). \end{aligned}$$

(10.31)

This concludes the proof of (10.18). $\square$

11 A Simple Application: Duality Formulas

As a simple application of the constructions presented above, we show how the non-polynomial scheme can be used to easily obtain various duality formulas which relate members of the q-Askey scheme.

11.1 Duality Formula for the Askey–Wilson Polynomials

The duality formula (B.7) for the Askey–Wilson polynomials is an easy consequence of Theorem 3.2 and the self-duality property (3.5) of the function $\mathcal {R}$ defined in (3.1). Indeed, suppose Assumptions 1.1 and 3.1 are satisfied and define $\{\sigma _t^{(n)}\}_{n=0}^\infty \subset \mathbb {C}$ by

$$\begin{aligned} \sigma _t^{(n)} = \tfrac{iQ}{2}+\theta _1+\theta _t+ibn. \end{aligned}$$

(11.1)

Under the parameter correspondence

$$\begin{aligned}{} & {} {\tilde{\alpha }}_R = e^{2 \pi b \left( \frac{i Q}{2}+\theta _0+\theta _t\right) }, \quad {\tilde{\beta }}_R =e^{2 \pi b \left( \frac{i Q}{2}+\theta _1-\theta _\infty \right) }, \quad {\tilde{\gamma }}_R=e^{2 \pi b \left( \frac{i Q}{2}-\theta _0+\theta _t\right) }, \\{} & {} {\tilde{\delta }}_R=e^{2 \pi b \left( \frac{i Q}{2}+\theta _1+\theta _\infty \right) }, \quad q=e^{2i\pi b^2}, \end{aligned}$$

the function $\mathcal {R}$ satisfies, for each integer $n \ge 0$,

$$\begin{aligned} \lim \limits _{\sigma _t \rightarrow \sigma _t^{(n)}} \mathcal {R} \left[ \begin{array}{c} \theta _1\;\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\;\theta _0 \end{array};\begin{array}{c} \sigma _s \vspace{0.15cm} \\ \sigma _t \end{array},b\right] = R_n(e^{2\pi b \sigma _s};{\tilde{\alpha }}_R,{\tilde{\beta }}_R,{\tilde{\gamma }}_R,{\tilde{\delta }}_R,q), \end{aligned}$$

(11.2)

where $R_n$ are the Askey–Wilson polynomials defined in (B.1). Moreover, evaluating $\mathcal {R}$ at $\sigma _s = \sigma _s^{(n)}$ and $\sigma _t = \sigma _t^{(m)}$ with $n,m \in \mathbb {Z}_{\ge 0}$ and using (3.13) and (11.2), we obtain

$$\begin{aligned}{} & {} \mathcal {R} \left[ \begin{array}{c} \theta _1\;\;\;\theta _t\vspace{0.1cm}\\ \theta _{\infty }\;\;\theta _0 \end{array};\begin{array}{c} \sigma _s^{(n)} \vspace{0.15cm} \\ \sigma _t^{(m)} \end{array},b\right] = R_n(e^{2\pi b \sigma _t^{(m)}};\alpha _R,\beta _R,\gamma _R,\delta _R,q) \nonumber \\{} & {} \quad = R_m(e^{2\pi b \sigma _s^{(n)}};{\tilde{\alpha }}_R,{\tilde{\beta }}_R,{\tilde{\gamma }}_R,{\tilde{\delta }}_R,q). \end{aligned}$$

(11.3)

Employing the symmetry $R_n(z;\alpha ,\beta ,\gamma ,\delta ,q) =R_n(z^{-1};\alpha ,\beta ,\gamma ,\delta ,q)$ and observing that the parameters in (3.12) satisfy

$$\begin{aligned} \alpha _R^2 = \frac{{\tilde{\alpha }}_R {\tilde{\beta }}_R {\tilde{\gamma }}_R {\tilde{\delta }}_R}{q}, \qquad \beta _R = \frac{{\tilde{\alpha }}_R {\tilde{\beta }}_R}{\alpha _R}, \qquad \gamma _R = \frac{{\tilde{\alpha }}_R {\tilde{\gamma }}_R}{\alpha _R}, \qquad \delta _R = \frac{{\tilde{\alpha }}_R {\tilde{\delta }}_R}{\alpha _R},\nonumber \\ \end{aligned}$$

(11.4)

and that

$$\begin{aligned} e^{-2\pi b\sigma _s^{(n)}} = \alpha _R^{-1} q^{-n}, \qquad e^{-2\pi b\sigma _t^{(m)}} = {\tilde{\alpha }}_R^{-1} q^{-m}, \end{aligned}$$

(11.5)

we recover the duality formula (B.7) for the Askey–Wilson polynomials.

11.2 Duality Formula Relating the Continuous Dual q-Hahn and the big q-Jacobi Polynomials

Suppose that Assumptions 1.1 and 4.6 are satisfied. Evaluating the function $\mathcal {H}$ at $\nu =\nu _n$ and $\sigma _s = \sigma _s^{(m)}$ with $n,m \in \mathbb {Z}_{\ge 0}$ and utilizing the polynomial limits (4.39) and (4.47), we obtain a duality formula relating the polynomials $H_n$ and $J_n$:

$$\begin{aligned}{} & {} \mathcal {H}(b,\theta _0,\theta _t,\theta _*,\sigma _s^{(m)},\nu _n) = H_n(e^{2\pi b \sigma _s^{(m)}};\alpha _H,\beta _{H},\gamma _{H},q)\nonumber \\{} & {} \quad = J_m(q^{-n};\alpha _J,\beta _J,\gamma _J;q). \end{aligned}$$

(11.6)

Employing the symmetry $H_n(z; \alpha ,\beta ,\gamma ,q)=H_n(z^{-1}; \alpha ,\beta ,\gamma ,q)$ and observing that the parameters in (4.38) and (4.46) are related by

$$\begin{aligned} \alpha _J = \frac{\alpha _H \beta _H}{q}, \qquad \beta _J = \frac{\alpha _H}{\beta _H}, \qquad \gamma _J = \frac{\alpha _H \gamma _H}{q}, \qquad e^{-2\pi b \sigma _s^{(m)}} = \alpha _H^{-1} q^{-m},\nonumber \\ \end{aligned}$$

(11.7)

we recognize the duality formula (B.16).

Remark 11.1

It is natural to ask whether there exists a well-defined limit of the duality formula (3.5) for $\mathcal R$, such that (11.6) follows directly by polynomial specialisation of the resulting duality formula for $\mathcal H$. However, we expect that $\mathcal H$ does not satisfy a duality relation analogous to (3.5) exchanging the parameters $\sigma _s$ and $\nu$. One apparent obstruction for the existence of such a duality formula is that, unlike the $\mathcal R$-function, the two dual difference operators $\tilde{D}_{\mathcal {H}}(b,\nu )$ and $D_{\mathcal {H}}(b,\sigma _s)$ defined in (4.30) and (4.23), respectively, possess different analytic properties. More precisely, the coefficients of $\tilde{D}_{\mathcal {H}}(b,\nu )$ are entire functions of $\nu$, whereas the coefficients of $D_{\mathcal {H}}(b,\sigma _s)$ are meromorphic functions of $\sigma _s$. If our expectation is true, it means that all other families in the non-polynomial scheme also do not exhibit a duality formula of this kind.

11.3 Duality Formula Relating the Little q-Jacobi and the Al-Salam Chihara Polynomials

Suppose that Assumptions 1.1 and 5.5 are satisfied. Evaluating $\mathcal {S}(b,\theta _0,\theta _t,\sigma _s,\rho )$ at $\rho =\rho _n$ and $\sigma _s=\sigma _s^{(m)}$ with $n,m \in \mathbb {Z}_{\ge 0}$ and utilizing the limits (5.33) and (5.35), we obtain a duality formula relating the polynomials $S_n$ and $Y_n$:

$$\begin{aligned} S_n(e^{2\pi b \sigma _s^{(m)}};\alpha _S,\beta _S,q) = Y_m(q^{-n}; \alpha _Y,\beta _Y,q), \qquad n,m=0,1,2,\dots . \end{aligned}$$

(11.8)

Employing the symmetry $S_n(z;\alpha _S,\beta _S,q)=S_n(z^{-1};\alpha _S,\beta _S,q)$ and observing that

$$\begin{aligned} \alpha _Y = \frac{\alpha _S \beta _S}{q}, \qquad \beta _Y = \frac{\alpha _S}{\beta _S}, \qquad e^{-2\pi b \sigma _s^{(m)}} = \alpha _S^{-1}q^{-m}, \end{aligned}$$

(11.9)

we recover the duality formula (B.28).

Appendices

Appendix A: q-Hypergeometric Series

The q-hypergeometric series $_{s+1}\phi _s$ is defined by

$$\begin{aligned} _{s+1}\phi _s\left[ \begin{matrix} a_1, \dots , a_{s+1} \\ b_1, \dots , b_s \end{matrix} ;q,z\right] = \sum _{k=0}^\infty \frac{(a_1, \dots ,a_{s+1};q)_k}{(b_1, \dots ,b_s,q;q)_k}z^k, \end{aligned}$$

(A.1)

where the q-Pochammer symbols $(a;q)_n$ and $(a_1,a_2,\dots ,a_m;q)_n$ are given by

$$\begin{aligned} (a;q)_n = \prod _{k=0}^{n-1} (1-a q^k) \quad \text {and} \quad (a_1,a_2,\dots ,a_m;q)_n = \prod _{j=1}^m (a_j;q)_n. \end{aligned}$$

(A.2)

Note that the sum in (A.1) contains only finitely many terms if one of the $a_i$ in the numerator is equal to $q^{-n}$ for some integer $n\ge 1$. Otherwise, the sum converges for $|z|<1$.

Appendix B: The q-Askey Scheme

1.1 B.1 Askey–Wilson Polynomials

The Askey–Wilson polynomials $R_n$, defined by

$$\begin{aligned} R_n(z;\alpha ,\beta ,\gamma ,\delta ,q) = {}_4 \phi _3\left( \left. \begin{aligned}\begin{matrix} q^{-n}, \alpha \beta \gamma \delta q^{n-1}, \alpha z, \alpha z^{-1} \\ \alpha \beta , \alpha \gamma , \alpha \delta \end{matrix} \end{aligned}\right| q;q \right) , \end{aligned}$$

(B.1)

are the most general polynomials of the q-Askey scheme. The normalization in (B.1) for $R_n$ is related to the standard normalization of [25, Eq. (14.1.1)] by

$$\begin{aligned} p_n\left( \tfrac{z+z^{-1}}{2};\alpha ,\beta ,\gamma ,\delta ,q\right) = \alpha ^{-n} (\alpha \beta ,\alpha \gamma ,\alpha \delta ;q)_n ~ R_n(z;\alpha ,\beta ,\gamma ,\delta ,q). \end{aligned}$$

(B.2)

Since $p_n(x;\alpha ,\beta ,\gamma ,\delta ,q)$ is a polynomial of order n in x, $R_n$ is a polynomial of order n in $z + z^{-1}$. The polynomials $R_n$ satisfy the three-term recurrence relation

$$\begin{aligned} (L_R R_n)(z;\alpha ,\beta ,\gamma ,\delta ,q) = (z+z^{-1}) R_n(z;\alpha ,\beta ,\gamma ,\delta ,q), \end{aligned}$$

(B.3)

where the operator $L_R$ is given by

$$\begin{aligned} L_R = a^{+}_n T^+ + (\alpha +\alpha ^{-1}-a^{+}_n-a^{-}_n) + a^{-}_n T^-, \end{aligned}$$

(B.4)

with $T^{\pm }p_n(x):= p_{n \pm 1}(x)$ and

$$\begin{aligned} \begin{aligned}&a^{+}_n = \frac{\left( 1-\alpha \beta q^n\right) \left( 1-\alpha \gamma q^n\right) \left( 1-\alpha \delta q^n\right) \left( 1-\alpha \beta \gamma \delta q^{n-1}\right) }{\alpha \left( 1-\alpha \beta \gamma \delta q^{2n-1}\right) \left( 1-\alpha \beta \gamma \delta q^{2n}\right) }, \\&a^{-}_n = \frac{\alpha \left( 1-q^n\right) \left( 1-\beta \gamma q^{n-1}\right) \left( 1-\beta \delta q^{n-1}\right) \left( 1-\gamma \delta q^{n-1}\right) }{\left( 1-\alpha \beta \gamma \delta q^{2n-2}\right) \left( 1-\alpha \beta \gamma \delta q^{2n-1}\right) }. \end{aligned}\end{aligned}$$

(B.5)

The Askey–Wilson polynomials also possess a symmetry exchanging the parameters n and z [36]. More precisely, define dual parameters ${\tilde{\alpha }},{\tilde{\beta }},{\tilde{\gamma }},{\tilde{\delta }}$ such that

$$\begin{aligned} \alpha ^2 = q^{-1} {\tilde{\alpha }} {\tilde{\beta }} {\tilde{\gamma }} {\tilde{\delta }}, \qquad \beta = \frac{{\tilde{\alpha }} {\tilde{\beta }}}{\alpha }, \qquad \gamma = \frac{{\tilde{\alpha }} {\tilde{\gamma }}}{\alpha }, \qquad \delta =\frac{{\tilde{\alpha }} {\tilde{\delta }}}{\alpha }. \end{aligned}$$

(B.6)

Then, by [36, Eq. (27)],

$$\begin{aligned} R_n(\alpha ^{-1} q^{-m};\alpha ,\beta ,\gamma ,\delta ,q) = R_m({\tilde{\alpha }}^{-1} q^{-n};{\tilde{\alpha }},{\tilde{\beta }},{\tilde{\gamma }},{\tilde{\delta }},q),\qquad n,m=0,1,2,\dots .\nonumber \\ \end{aligned}$$

(B.7)

1.2 B.2 Continuous dual q-Hahn Polynomials

The continuous dual q-Hahn polynomials $H_n(z;\alpha ,\beta ,\gamma ,q)$ are defined by

$$\begin{aligned} H_n(z;\alpha ,\beta ,\gamma ,q) = R_n(z;\alpha ,\beta ,\gamma ,0,q)= {}_3\phi _2\left( \left. \begin{matrix} q^{-n},\alpha z,\alpha z^{-1} \\ \alpha \beta , \alpha \gamma \end{matrix}\right| q;q\right) . \end{aligned}$$

(B.8)

They satisfy the three-term recurrence relation

$$\begin{aligned} \left( L_H H_n\right) (z;\alpha ,\beta ,\gamma ,q)=(z+z^{-1})~H_n(z;\alpha ,\beta ,\gamma ,q), \end{aligned}$$

(B.9)

where the operator $L_H$ is defined by

$$\begin{aligned} L_H = b^{+}_n T^++\left( \alpha +\alpha ^{-1}-b^{+}_n-b^{-}_n\right) +b^{-}_n T^-, \end{aligned}$$

(B.10)

with

$$\begin{aligned} b^{+}_n=\alpha ^{-1}(1-\alpha \beta q^n)(1-\alpha \gamma q^n), \qquad b^{-}_n=\alpha (1-q^n)(1-\beta \gamma q^{n-1}). \end{aligned}$$

(B.11)

1.3 B.3 Big q-Jacobi Polynomials

The big q-Jacobi polynomials $J_n(x;\alpha ,\beta ,\gamma ;q)$ are defined by

$$\begin{aligned}{} & {} J_n(x;\alpha ,\beta ,\gamma ;q) = \lim \limits _{\lambda \rightarrow 0} R_n\left( \frac{x}{\lambda };\lambda ,\frac{\alpha q}{\lambda },\frac{\gamma q}{\lambda },\frac{\lambda \beta }{\gamma },q\right) \nonumber \\{} & {} \quad = {}_3\phi _2\left( \left. \begin{aligned} \begin{matrix} q^{-n},\alpha \beta q^{n+1}, x \\ \alpha q, \gamma q\end{matrix} \end{aligned} \right| q;q\right) . \end{aligned}$$

(B.12)

They satisfy the three-term recurrence relation

$$\begin{aligned} L_J J_n(x;\alpha ,\beta ,\gamma ;q)=x J_n(x;\alpha ,\beta ,\gamma ;q), \end{aligned}$$

(B.13)

where the operator $L_J$ is defined by

$$\begin{aligned} L_J = c^{+}_n T^+ + \left( 1-c^{+}_n-c^{-}_n\right) + c^{-}_n T^-, \end{aligned}$$

(B.14)

with

$$\begin{aligned} \left\{ \begin{aligned} c^{+}_n&= \frac{\left( 1-\alpha q^{n+1}\right) \left( 1-\alpha \beta q^{n+1}\right) \left( 1-\gamma q^{n+1}\right) }{\left( 1-\alpha \beta q^{2n+1}\right) \left( 1-\alpha \beta q^{2n+2}\right) }, \\ c^{-}_n&= -\alpha \gamma q^{n+1} \frac{\left( 1-q^n\right) \left( 1-\alpha \beta \gamma ^{-1} q^n\right) \left( 1-\beta q^n\right) }{\left( 1-\alpha \beta q^{2n}\right) \left( 1-\alpha \beta q^{2n+1}\right) }. \end{aligned}\right. \end{aligned}$$

(B.15)

There exists a duality between the big q-Jacobi and the continuous dual q-Hahn polynomials which is inherited from the duality (B.7) of the Askey–Wilson polynomials. More precisely, by [36, Eq. (44)],

$$\begin{aligned} H_n\left( \alpha ^{-1} q^{-m}; \alpha ,\beta ,\gamma ,q\right){} & {} = J_m\left( q^{-n}; q^{-1} \alpha \beta , \alpha \beta ^{-1}, q^{-1} \alpha \gamma ; q\right) ,\nonumber \\{} & {} \qquad n,m=0,1,2,\dots . \end{aligned}$$

(B.16)

1.4 B.4 Al-Salam–Chihara Polynomials

The Al-Salam Chihara polynomials $S_n(z;\alpha ,\beta ,q)$ are defined by

$$\begin{aligned}{} & {} S_n(z;\alpha ,\beta ,q) = H_n(z;\alpha ,\beta ,0,q)\nonumber \\{} & {} \quad = \frac{\alpha ^n}{\left( \alpha \beta ;q \right) _n} \left( \alpha z;q \right) _n z^{-n} {}_2\phi _1\left( \left. \begin{aligned} \begin{matrix} q^{-n}, \beta z^{-1} \\ \alpha ^{-1} q^{1-n} z^{-1} \end{matrix} \end{aligned} \right| q;\alpha ^{-1} q z \right) , \end{aligned}$$

(B.17)

where $H_n$ are the continuous dual q-Hahn polynomials defined in (B.8). In (B.17), we use the normalization of [36, Eq. (59)]. The polynomials $S_n$ satisfy the three-term recurrence relation

$$\begin{aligned} L_S S_n(z;\alpha ,\beta ,q) = \left( z+z^{-1}\right) S_n(z;\alpha ,\beta ,q), \end{aligned}$$

(B.18)

where the operator $L_S$ is defined by

$$\begin{aligned} L_S = \left( \alpha ^{-1}-\beta q^n\right) T^+ +\left( \alpha +\beta \right) q^n+ \alpha \left( 1-q^n \right) T^-. \end{aligned}$$

(B.19)

1.5 B.5 Little q-Jacobi Polynomials

The Little q-Jacobi polynomials $p_n(x;\alpha ,\beta ,q)$ are defined by (see [36, Eq. (66)])

$$\begin{aligned} p_n(x;\alpha ,\beta ,q) = {}_2\phi _1\left( \left. \begin{aligned} \begin{matrix} q^{-n}, \alpha \beta q^{n+1} \\ \alpha q \end{matrix} \end{aligned} \right| q;qx\right) , \end{aligned}$$

(B.20)

or, equivalently, by [35, Eq. (3.38)]

$$\begin{aligned} p_n(x;\alpha ,\beta ,q) = (-q\beta )^{-n} q^{-\frac{n(n-1)}{2}} \frac{(q\beta ;q)_n}{(q \alpha ;q)_n} {}_3\phi _2\left( \left. \begin{aligned} \begin{matrix} q^{-n}, q^{n+1}\alpha \beta , q \beta x \\ q \beta , 0 \end{matrix} \end{aligned} \right| q;q\right) .\nonumber \\ \end{aligned}$$

(B.21)

They arise as limits of the big q-Jacobi polynomials in two ways. First, we have (see [36, Eq. (68)])

$$\begin{aligned} p_n(x;\alpha ,\beta ,q) = \lim \limits _{\gamma \rightarrow \infty } J_n(\gamma q x;\alpha ,\beta ,\gamma ;q). \end{aligned}$$

(B.22)

Second, we have

$$\begin{aligned} p_n(x;\alpha ,\beta ,q) = (-q \beta )^{-n} q^{-\frac{n(n-1)}{2}} \frac{(q\beta ;q)_n}{(q \alpha ;q)_n} Y_n(q \beta x; \beta , \alpha , q), \end{aligned}$$

(B.23)

where

$$\begin{aligned} Y_n(x;\alpha , \beta , q) = \lim \limits _{\gamma \rightarrow 0} J_n(x;\alpha ,\beta ,\gamma ,q) = {}_3\phi _2\left( \left. \begin{aligned} \begin{matrix} q^{-n}, q^{n+1}\alpha \beta , x \\ \alpha q, 0 \end{matrix} \end{aligned} \right| q;q\right) . \end{aligned}$$

(B.24)

The polynomials $Y_n$ satisfy the three-term recurrence relation (see [36, Eqs. (73)–(74)])

$$\begin{aligned} L_Y Y_n(x;\alpha ,\beta ,q) = x Y_n(x;\alpha ,\beta ,q), \end{aligned}$$

(B.25)

where the operator $L_Y$ is defined by

$$\begin{aligned} L_Y = y^+_n T^+ + 1-y^+_n-y^-_n + y^-_n T^-, \end{aligned}$$

(B.26)

with

$$\begin{aligned}&y^+_n = \frac{\left( 1-\alpha q^{n+1}\right) \left( 1-\alpha \beta q^{n+1} \right) }{\left( 1-\alpha \beta q^{2n+1} \right) \left( 1-\alpha \beta q^{2n+2} \right) }, \nonumber \\&y^-_n = q^{2n+1} \alpha ^2 \beta \frac{\left( 1-q^n \right) \left( 1-\beta q^n \right) }{\left( 1-\alpha \beta q^{2n} \right) \left( 1-\alpha \beta q^{2n+1} \right) }. \end{aligned}$$

(B.27)

Finally, there exists a duality between the little q-Jacobi polynomials and the Al-Salam Chihara polynomials which is inherited from (B.16). More precisely, from [36, Eq. (75)] we have

$$\begin{aligned} S_n\left( \alpha ^{-1} q^{-m}; \alpha ,\beta ,q \right) = Y_m \left( q^{-n}; q^{-n} \alpha \beta , \alpha \beta ^{-1}, q\right) , \qquad n,m=0,1,2,\dots . \nonumber \\\end{aligned}$$

(B.28)

1.6 B.6 Big q-Laguerre Polynomials

The big q-Laguerre polynomials $L_n(x;\alpha ,\beta ;q)$ are defined by setting $\beta =0$ in the big q-Jacobi polynomials (see [25, Eq. (14.11.1)]):

$$\begin{aligned}{} & {} L_n(x;\alpha ,\beta ;q)=J_n(x;\alpha ,0,\beta ;q) \nonumber \\{} & {} \quad =\frac{1}{\left( \beta ^{-1} q^{-n};q \right) _n}{}_2\phi _1\left( \left. \begin{aligned} \begin{matrix} q^{-n}, \alpha q x^{-1} \\ \alpha q \end{matrix} \end{aligned} \right| q;x \beta ^{-1}\right) . \end{aligned}$$

(B.29)

The polynomials $L_n$ satisfy the three-term recurrence relation

$$\begin{aligned} L_L L_n(x;\alpha ,\beta ,q) = x \; L_n(x;\alpha ,\beta ,q), \end{aligned}$$

(B.30)

where the operator $L_L$ is defined by

$$\begin{aligned} L_L = l^+_n T^+ +(1-l^+_n-l^-_n) + l^-_n T^-, \end{aligned}$$

(B.31)

with

$$\begin{aligned}&l_n^+ = \left( 1-\alpha q^{n+1} \right) \left( 1-\beta q^{n+1} \right) , \qquad l_n^- = -\alpha \beta q^{n+1} \left( 1-q^n\right) . \end{aligned}$$

(B.32)

1.7 B.7 Little q-Laguerre Polynomials

The little q-Laguerre (or Wall) polynomials $W_n(x;\alpha ,q)$ are obtained from the big q-Laguerre polynomials as follows:

$$\begin{aligned} W_n(x;\alpha ,q) = \lim \limits _{\beta \rightarrow -\infty } L_n(\beta q x;\alpha ,\beta ;q) = {}_2\phi _1\left( \left. \begin{aligned} \begin{matrix} q^{-n}, 0 \\ \alpha q \end{matrix} \end{aligned} \right| q;q x\right) . \end{aligned}$$

(B.33)

They satisfy the recurrence relation

$$\begin{aligned} L_W W_n(x;\alpha ,q) = -x W_n(x;\alpha ,q), \end{aligned}$$

(B.34)

where the operator $L_W$ is defined by

$$\begin{aligned} L_W = w^+_n T^+ -(w^+_n+w^-_n) + w^-_n T^-, \end{aligned}$$

(B.35)

with

$$\begin{aligned}&w_n^+ = q^n \left( 1-\alpha q^{n+1} \right) , \qquad w_n^- = \alpha q^n \left( 1-q^n \right) . \end{aligned}$$

(B.36)

1.8 B.8 Continuous Big q-Hermite Polynomials

The continuous big q-Hermite polynomials $X_n(z;\alpha ,q)$ are defined by

$$\begin{aligned} X_n(z;\alpha ,q)=\lim \limits _{\beta \rightarrow 0} S_n(z;\alpha ,\beta ,q) = \alpha ^n z^n {}_2\phi _0\left( \left. \begin{aligned} \begin{matrix} q^{-n}, \alpha z \\ - \end{matrix} \end{aligned} \right| q;q^n z^{-2} \right) , \end{aligned}$$

(B.37)

where $S_n$ are the Al-Salam-Chihara polynomials defined in (B.17). The normalization in (B.37) is obtained by multiplying $H_n(\tfrac{1}{2}(z+z^{-1}),\alpha |q)$ in [25, Eq. (14.18.1)] by $\alpha ^n$. They satisfy the recurrence relation

$$\begin{aligned} R_{X_n} X_n(z;\alpha ,q) = \left( z+z^{-1}\right) X_n(z;\alpha ,q), \end{aligned}$$

(B.38)

where the operator $R_{X_n}$ is defined by

$$\begin{aligned} R_{X_n} = \alpha ^{-1} T^+ + \alpha q^n + \alpha \left( 1-q^n \right) T^-. \end{aligned}$$

(B.39)

1.9 B.9 Continuous q-Hermite Polynomials

The continuous q-Hermite polynomials $Q_n(z;q)$ are defined by

$$\begin{aligned} Q_n(z;q) = \lim \limits _{\alpha \rightarrow 0} \alpha ^{-n} X_n(z;\alpha ,q)= z^n {}_2\phi _0\left( \left. \begin{aligned} \begin{matrix} q^{-n}, 0 \\ - \end{matrix} \end{aligned} \right| q;q^n z^{-2} \right) . \end{aligned}$$

(B.40)

They satisfy the recurrence relation

$$\begin{aligned} L_Q Q_n(z;q) = \left( z+z^{-1}\right) Q_n(z;q), \end{aligned}$$

(B.41)

where the operator $L_Q$ is defined by

$$\begin{aligned} L_Q = T^++\left( 1-q^n \right) T^-. \end{aligned}$$

(B.42)