Non-polynomial $q$-Askey scheme: integral representations, eigenfunction properties, and polynomial limits

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 1.1.Introduction.The Askey scheme is a way of organizing orthogonal polynomials.Each element of the scheme represents a family of polynomials, and the families at a given level can be obtained from those at the level above by taking suitable limits [2].The Wilson and Racah polynomials lie at the top level and are thus the most general polynomials of the scheme.In addition to the classical Jacobi, Laguerre, and Hermite polynomials, the Askey scheme also includes less well-known families such as the continuous dual Hahn and Meixner-Pollaczek polynomials.
The q-Askey scheme is a generalization of the Askey scheme.The polynomials in the q-Askey scheme are q-analogs of the polynomials in the Askey scheme, and the elements in the latter scheme can be recovered by taking the limit q → 1 [17].The Askey-Wilson and q-Racah polynomials lie at the top level and all the other polynomials in the q-Askey scheme can be obtained from these via appropriate limit operations.We refer to Appendix B for a list of basic properties of elements in the q-Askey scheme.
Our goal in this paper is to present a non-polynomial generalization of the 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 Figure 1.Just like the Askey and q-Askey schemes, the non-polynomial scheme we present has five levels, see Figure 2. Whereas each element in the q-Askey scheme is a family of polynomials, each element of the nonpolynomial scheme is a meromorphic function.These meromorphic functions are given by contour integrals whose integrands are built from Ruijsenaars' hyperbolic gamma function [38] (alternatively, the integrands can be expressed in terms of Faddeev's quantum dilogarithm [10], Woronowicz's quantum exponential [47], or Kurokawa's double sine function [25]).The non-polynomial scheme is a generalization of the q-Askey scheme in the sense that each element in Figure 1 is obtained from an element at the same level in Figure 2 when one of the variables is suitably discretized.
A crucial property of the elements of the q-Askey scheme is that they are joint eigenfunctions.More precisely, if p n (x) is a family of orthogonal polynomials in the q-Askey scheme, then the p n are eigenfunctions of a second-order difference operator in the n-variable as well as of a second-order q-difference operator in the x-variable.We refer to the corresponding eigenfunction equations as the recurrence and difference equations Big q-Jacobi Al-Salam Chihara Continuous big q-Hermite Continuous q-Hermite Little q-Jacobi Big q-Laguerre Little q-Laguerre Little q-Laguerre with α = 0 H(b, θ 0 , θ t , θ * , σ s , ν) S(b, θ 0 , θ t , σ s , ρ) L(b, θ t , θ, λ, µ) W(b, θ t , κ, ω) X (b, θ, σ s , ω) Q(b, σ s , η) Figure 2. The elements of the non-polynomial scheme which generalize the ones in Figure 1.
for p n , respectively.The elements of the non-polynomial scheme are also joint eigenfunctions.Indeed, we will show that each 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 difference equation.
The first (top) level of the non-polynomial scheme consists of a single function, namely, Ruijsenaars' hypergeometric function R1 [39].Below this top level, there are four further levels involving functions which can be obtained from R by taking various limits; we denote these functions by H, S, X , Q, L, W, and M, where the letters are chosen so that H is a non-polynomial generalization of the continuous dual q-Hahn polynomials which are denoted by H n in Appendix B, S is a non-polynomial generalization of the Al-Salam Chihara polynomials which are denoted by S n in Appendix B, etc.
Each of the functions in the non-polynomial scheme depends on a number of parameters as well as two variables.For example, H depends on the parameters b, θ 0 , θ t , θ * as well as the two variables σ s , ν, while S depends on the parameters b, θ 0 , θ t as well as the two variables σ s , ρ, see Figure 2. As a matter of notation, we will refer to the top level of the scheme (which involves R) as the first level, to the next level (which involves 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 H and S, possess two different polynomial limits: H reduces to the continuous dual q-Hahn polynomials when ν is discretized and to the big q-Jacobi polynomials when σ s is discretized; similarly, S reduces to the Al-Salam Chihara polynomials when ρ is discretized and to the little q-Jacobi polynomials when σ 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 [7,[44][45][46], in representation theory of quantum algebras [1,3,18,19,21,22,30,31,48], and in the geometry of Painlevé equations [29].We expect the functions in the proposed non-polynomial scheme to also be relevant in many different contexts (see Section 12 for a related discussion).This is certainly true of the top element, Ruijsenaars' R-function.As an example of the broad relevance of R, we note that one of the authors recently showed [36] that R is equivalent (up to a change of variables) to the Virasoro fusion kernel which is a central object in conformal field theory [6,[33][34][35].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 [26], we introduced a family of confluent Virasoro fusion kernels C k while studying confluent conformal blocks of the second kind of the Virasoro algebra.Later, we realized that the 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 [27].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' R-function as our starting point.The result of [36] implies that these two choices are equivalent, but we have found that the scheme originating from R is simpler and mathematically more convenient.
1.2.Organization of the paper.In Section 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 R, H, S, X , Q, L, W, M that make up the non-polynomial scheme are considered one by one in the eight Sections 3-10.In Section 11, we derive-as an easy application of the non-polynomial scheme-a few duality formulas that relate members of the q-Askey scheme.Section 12 presents some perspectives.In the two appendices, we have collected relevant definitions and properties of q-hypergeometric series and of the q-Askey scheme.1.3.Standing assumption.Throughout the paper, we make the following assumption.Assumption 1.1 (Restrictions on the parameters).We assume that (b, θ ∞ , θ 1 , θ t , θ 0 , θ * , θ) ∈ (0, ∞) × R 6 .
Acknowledgements.The authors acknowledge support from the European Research Council, Grant Agreement No. 682537, the Swedish Research Council, Grant No. 2015-05430, and the Ruth and Nils-Erik Stenbäck Foundation.

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 where The function s b (z) is closely related to Ruijsenaars' hyperbolic gamma function [38], Faddeev's quantum dilogarithm function [10], Woronowicz's quantum exponential function [47], and Kurokawa's double sine function [25].More precisely, it is related to Ruijsenaars' hyperbolic gamma function G in [39, Eq. (A. 3)] by It follows from (2.2) and the results in [39] that s b (z) is a meromorphic function of z ∈ C with zeros {z m,l } ∞ m,l=0 and poles {p m,l } ∞ m,l=0 located at , (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 ) ∈ Z ≥0 × Z ≥0 such that z mi,li = 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 Furthermore, s b is a meromorphic solution of the following pair of difference equations: Applying the difference equations (2.5) recursively, it can be verified that, for any integer m ≥ 0, where the q-Pochhammer symbol (a; q) m is defined in Appendix A. Finally, the function s b (z) has the obvious symmetry and possesses an asymptotic formula which is a consequence of [39, Theorem A.1] and (2.2):For each ǫ > 0, uniformly for (b, Im z) in compact subsets of (0, ∞) × R.

The function R
The top element of the non-polynomial q-Askey scheme presented in this paper is Ruijsenaars' hypergeometric function R [39].Using the notation of [36], this function can be expressed as where the prefactor P R is given by and the integrand I R is defined by In view of (2.3), the integrand I 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 = −θ 0 − θ t ± σ s − iQ 2 and x = −θ 1 − θ t ± σ t − iQ 2 , and four vertical upward sequences starting at x = 0, x = −2θ t , and x = −θ 0 − θ 1 ± θ ∞ − θ t .The contour R in (3.1) is any curve from −∞ to +∞ which separates the four upward from the four downward sequences of poles.If in addition to Assumption 1.1, we also assume that σ s , σ t ∈ R, then the contour of integration R can be chosen to be any curve from −∞ to +∞ lying in the open strip Im x ∈ (−Q/2, 0).
Thanks to the symmetry (2.7) of the function s b , we have where and For (σ s , σ t ) ∈ C 2 , the function R satisfies the following four difference equations [39] (using the notation of [36]): 3.2.Polynomial limit.It was shown in [39] that the function R reduces to the Askey-Wilson polynomials when one of the variables σ s and σ t is suitably discretized.This result was reobtained in the CFT setting in [27].We now recall the result of [27].In addition to Assumption 1.1, we need the following assumption.

The function 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 R. The second level of the non-polynomial scheme involves the function H(b, θ 0 , θ t , θ * , σ s , ν), which is defined as the confluent limit Λ → −∞ of the top element R evaluated at In this section, we derive an integral representation for the function H and we show that it is a joint eigenfunction of four difference operators, two acting on σ s and the other two on ν.We also show that H reduces to the continuous dual q-Hahn and the big q-Jacobi polynomials when ν and σ s are suitably discretized, respectively.Since these polynomials lie at the second level of the q-Askey scheme, this shows that H indeed provides a natural non-polynomial generalization of the elements at the second level.
The next theorem shows that, for each choice of (b, θ 0 , θ t , θ * ) ∈ (0, ∞) × R 3 , H is a well-defined and analytic function of where ∆ where The theorem also provides an integral representation for H for (σ s , ν) satisfying (4.3).In fact, even if the requirement Im ν > −Q/2 is needed to ensure convergence of the integral in the integral representation for H, we will show later in this section, with the help of the difference equations satisfied by H, that H extends to a meromorphic function of (σ s , ν) ∈ C 2 .
Theorem 4.2.Suppose that Assumption 1.1 holds.Let ∆ H,σs , ∆ ν ⊂ C be the discrete subsets defined in (4.4).Then the limit in (4.2) exists uniformly for (σ s , ν) in compact subsets of Moreover, H is an analytic function of (σ s , ν) ∈ D H and admits the following integral representation: where the dependence of P H and I H on b, θ 0 , θ t , θ * is omitted for simplicity, ) and the contour H is any curve from −∞ to +∞ which separates the three upward from the three downward sequences of poles.In particular, H is a meromorphic function of (σ s , ν) , then the contour H can be any curve from −∞ to +∞ lying within the strip Im x ∈ (−Q/2, 0).
where the dependence of X(x, Λ) on b, θ 0 , θ t , θ * , σ s , ν is omitted for simplicity, and (4.9) Due to the properties (2.3) of the function s b , the function I H (•, σ s , ν) possesses three increasing sequences of poles starting at x = 0, x = −2θ t and x = −θ 0 − θ * − θ t , as well as three decreasing sequences of poles starting at The discrete sets ∆ H,σs and ∆ ν contain all the values of σ s and ν, 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 = − iQ 2 − θ * 2 − θ t + ν and the increasing sequence starting at x = 0. Poles from these two sequences collide if and only if , giving rise to the first set on the right-hand side of (4.4b).
Similarly, (2.3) implies that X(•, Λ) possesses one increasing sequence of poles starting at x = −Λ − θ 0 − θ t and one decreasing sequence of poles starting at x = − iQ 2 − Λ − θ * 2 − θ t − ν.The real parts of the poles in these two sequences tend to +∞ as Λ → −∞.The increasing sequence lies in the half-plane Im x ≥ 0 and the decreasing sequence lies in the half-plane Im x ≤ −Im ν − Q/2.
The sets ∆ H,σs and ∆ ν also contain all the values of σ s and ν at which the prefactor P H (σ s , ν) has poles.For example, P H has poles originating from the factor giving rise to the last set on the right-hand side of (4.4b).Let K σs be a compact subset of C \ ∆ H,σs and let K ν be a compact subset of {Im ν > −Q/2} \ ∆ ν .Suppose (σ s , ν) ∈ K σs × K ν .Then, the above discussion shows that it is possible to choose a contour H = H(σ s , ν) from −∞ to +∞ which separates the four upward from the four downward sequences of poles of X(•, Λ)I H (•, σ s , ν).It also follows that if we let the right tail of H approach the horizontal line Im x = −ǫ as Re x → +∞, where ǫ > 0 is sufficiently small, then there exists a N < 0 such that H can be chosen to be independent of Λ for Λ < −N .Thus, for such a choice of H, (3.1) and (4.8) imply that, for all (σ s , ν) ∈ K σs × K ν and all Λ < −N , uniformly for (σ s , ν) ∈ K σs × K ν and for x in bounded subsets of H.We deduce that lim uniformly for (σ s , ν) ∈ K σs × K ν and x in bounded subsets of H. Using the asymptotic formula (2.8) for s b with ǫ = 1/2, we find that I H obeys the estimate uniformly for (σ s , ν) ∈ K σs × K ν and Im x in compact subsets of R. Since the contour H stays a bounded distance away from the increasing and the decreasing pole sequences, we infer that there exists a constant uniformly for (σ s , ν) ∈ K σs × K ν .In particular, since K ν ⊂ {Im ν > −Q/2}, I H has exponential decay on the left and right tails of the contour H. Suppose we can show that there exist constants c > 0 and C > 0 such that uniformly for all Λ < −N , x ∈ H, and (σ s , ν) ∈ K σs ×K ν .Then it follows from (4.10), (4.12), and Lebesgue's dominated convergence theorem that the limit in (4.2) exists uniformly for (σ s , ν) ∈ K σs × K ν and is given by (4.5).Since K σs ⊂ C \ ∆ H,σs and K ν ⊂ C \ ∆ ν are arbitrary compact subsets, this proves that the limit in (4.2) exists uniformly for (σ s , ν) in compact subsets of D H and proves (4.5).Moreover, the analyticity of H as a function of (σ s , ν) ∈ D H follows from the analyticity of R together with the uniform convergence on compact subsets.Alternatively, the analyticity of H in D H can be inferred directly from the representation (4.5).Indeed, the possible poles of H lie at such values of (σ s , ν) at which either the prefactor P H has a pole or at which the contour of integration gets pinched between two poles of the integrand I H , and the definitions of ∆ H,σs and ∆ ν 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 ǫ = 1/2 implies that there exist constants C 2 , C 3 , C 4 > 0 such that the inequalities ) ) hold uniformly for (σ s , ν) ∈ K σs × K ν .Combining the above estimates, we infer that there exists a constant C 5 such that uniformly for (σ s , ν) ∈ K σs × K ν .The inequality (4.19) can be rewritten as follows: uniformly for (σ s , ν) , then (4.20) implies that |X| is uniformly bounded for all x ∈ H, Λ < −N , and (σ s , ν) ∈ K σs × K ν ; hence (4.15) follows from (4.14) in this case.On the other hand, if which shows (4.15) also in this case.This completes the proof.
Furthermore, thanks to the symmetry (2.7) of s b , we have 4.2.Difference equations.We now show that the four difference equations (3.9) satisfied by the function R survive in the confluent limit (4.2).This implies that the function H is a joint eigenfunction of four difference operators, two acting on σ s and the remaining two on ν.
We know from Theorem 4.2 that H is a well-defined meromorphic function of (σ s , ν) ∈ C×{Im ν > −Q/2}.The difference equations will first be derived as equalities between meromorphic functions defined on this limited domain.However, the difference equations in ν can then be used to show that: (i) the limit in (4.2) exists for all ν in the whole complex plane away from a discrete subset, (ii) H is in fact a meromorphic function of (σ s , ν) in all of C 2 , and (iii) the four difference equations hold as equalities between meromorphic functions on C 2 , see Proposition 4.5.

First pair of difference equations. Define the difference operator H
where with Proposition 4.3.For σ s ∈ C and Im ν > −Q/2, the function H defined by (4.2) satisfies the following pair of difference equations: Proof.The proof consists of taking the confluent limit (4.2) of the difference equation (3.9a).On the one hand, we have lim On the other hand, straightforward computations using asymptotics of hyperbolic functions show that the following limits hold: ; b, σs = H 0 H (b, σ s ), (4.28)where H + R and H 0 R are defined in (3.7) and (3.8), respectively.Therefore we obtain lim where H R is given in (3.6).By Theorem 4.
where H0 H is defined by H0 , the function H satisfies the following pair of difference equations: Proof.It is straightforward to verify that the following limits hold: where H + R and H 0 R are defined in (3.7) and (3.8), respectively.We obtain lim where H R is defined in (3.6)Using the difference equations (4.33), we can show that H extends to a meromorphic function of ν everywhere in the complex plane.More precisely, we have the following proposition.
Then there is a discrete subset ∆ ⊂ C such that the limit in (4.2) exists for all ν ∈ C \ ∆.Moreover, the function H defined by (4.2) is a meromorphic function of (σ s , ν) ∈ C 2 and the four difference equations (4.26) and (4.33) hold as equalities between meromorphic functions of (σ s , ν) , b , taking the confluent limit Λ → −∞, and using (4.2), (4.35), and Theorem 4.2, we conclude that the limit in (4.2) exists also for where ∆ 1 is a discrete set.By iteration, we conclude that the limit in (4.2) exists for all ν ∈ C \ ∆, where ∆ 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 ν ∈ C \ ∆. 4.3.First polynomial limit.In this subsection, we show that the function H reduces to the continuous dual q-Hahn polynomials when ν 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 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 2iπb 2 is not a root of unity. Define n=0 is a subset of the set ∆ ν of possible poles of H defined in (4.4b).The following theorem shows that H still has a finite limit as ν → ν n for each n ≥ 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 H has a simple zero at each ν 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 H to the continuous dual q-Hahn polynomials).Let σ s ∈ C \ ∆ H,σs and suppose that Assumptions 1.1 and 4.6 are satisfied.Under the parameter correspondence the function H defined in (4.5) satisfies, for each integer n ≥ 0, 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 ν → ν n in the integral representation (4.5) for 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 Section 10 for the function 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 Moreover, in the limit ν → ν 0 , the pole of the function −1 located at x = 0, pinching the contour H. Therefore, before taking the limit ν → ν 0 , we deform the contour H into a contour H ′ which passes below x 0 .We obtain Using the residue (2.4), a straightforward computation yields The right-hand side of (4.41) has a simple pole at ν = ν 0 due to the factor s b (ν − θ * 2 − θ t ) −1 .Moreover, in the limit ν → ν 0 the second term in (4.40) vanishes thanks to the zero of P H . Thus we obtain , it is straightforward to verify that the right-hand side equals 1; this proves (4.39).
For each integer n ≥ 0, let P n denote the left-hand side of (4.39): The same kind of contour deformation used to establish the case n = 0 shows that the limit in (4.43) exists for all n ≥ 0. To show that P n equals the continuous dual q-Hahn polynomials H n , we consider the limit ν → ν n of the difference equation (4.33a).Using the parameter correspondence (4.38), it is straightforward to verify that lim ν→νn HH (b, ν) = R Hn , (4.44) where the operators HH and R Hn are defined in (4.30) and (B.10), respectively.Hence taking the limit ν → ν n of the difference equation (4.33a), we see that P n satisfies where z = e 2πbσs .Thus the P n satisfy the same recurrence relation (B.9) as the continuous dual q-Hahn polynomials evaluated at z = e 2πbσs .Since we have already shown that P 0 = H 0 = 1, we infer that P n = H n for all n ≥ 0 by induction, where H n is evaluated at z = e 2πbσs .This completes the proof of (4.39).
4.4.Second polynomial limit.In this subsection, we show that H reduces to the big q-Jacobi polynomials when σ s is suitably discretized.
Theorem 4.8 (From H to the big q-Jacobi polynomials).Let ν ∈ {Im ν > −Q/2} \ ∆ ν and suppose that Assumptions 1.1 and 4.6 are satisfied.Under the parameter correspondence the function H defined in (4.5) satisfies, for each integer n ≥ 0, lim where σ ) 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 (σ s − θ 0 − θ t ) in (4.6) has a simple zero located at σ s = σ On the other hand, in the limit ) −1 at x = 0. Therefore, before taking the limit σ s → σ (0) s , we deform the contour H into a contour H ′ which passes below x 0 .We obtain (4.48) A straightforward computation using (2.4) shows that The right-hand side has a simple pole at σ s = σ (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 ≥ 1.Let The same kind of contour deformation used for n = 0 shows that the limit in (4.47) exists for all n ≥ 0.Moreover, under the parameter correspondence (4.46), we have lim where the operators H H and R Jn are defined in (4.23) and (B.14), respectively.Hence taking the limit of (4.26a) and recalling that x J = e 2πb(θt+ θ *

+ iQ
2 ) e −2πbν , we see that P n satisfies R Jn 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 ≥ 1 follows by induction.Remark 4.9.It is also possible to define a function It can be shown that the limit in (4.52) exists for (σ s , ν) ∈ D H .Moreover, due to the asymptotic formula (2.8) for s b , the only difference between H ′ and H resides in the sign of the phases in the representation (4.5).In fact, the following two limits hold: We expect that a similar phenomenon is present for all the elements of the non-polynomial scheme.For simplicity, we will only study one of the two representatives for each element.

The function S
In this section, we introduce the function S(b, θ 0 , θ t , σ s , ρ) which is one of the two elements at the third level of the non-polynomial scheme, see Figure 2. The function S is defined as a confluent limit of the function H.We show that 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 ρ and σ s are suitably discretized, respectively.
where the discrete subsets ∆ S,σs and ∆ ρ are given by Let H be defined by (4.2).The function S is defined for (σ s , ρ) ∈ D S by and is extended meromorphically to (σ s , ρ) ∈ C 2 .
The next theorem shows that S is a well-defined meromorphic function of (σ s , ρ) ∈ C 2 .
Theorem 5.2.Suppose that Assumption 1.1 is satisfied.The limit in (5.3) exists uniformly for (σ s , ρ) in compact subsets of D S .Moreover, the function S is an analytic function of (σ s , ρ) ∈ (C\∆ S,σs ) × (C\∆ ρ ) and admits the following integral representation: where the dependence of P S and I S on b, θ 0 , θ t is omitted for simplicity, and the contour S is any curve from −∞ to +∞ which separates the three decreasing from the two increasing sequences of poles, with the requirement that its right tail satisfies for some δ > 0. In particular, S is a meromorphic function of (σ s , ρ) ∈ C 2 .If (σ s , ρ) ∈ R 2 , then the contour S can be any curve from −∞ to +∞ lying within the strip 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, θ 0 , θ t ) ∈ (0, ∞) × R 2 .It can be verified that where 2 e iπx(θ0+θ * +θt+ iQ 2 ) (5.9) Due to the properties (2.3) of the function s b , the function I S (•, σ s , ρ) possesses two increasing sequences of poles starting at x = 0 and x = −2θ t , as well as three decreasing sequences of poles starting at The discrete sets ∆ S,σs and ∆ ρ contain all the values of σ s and ρ, respectively, for which poles in any of the two increasing sequences collide with poles in any of the three decreasing sequences.The sets ∆ S,σs and ∆ ρ also contain all the values of σ s and ρ at which the prefactor P S (σ s , ρ) has poles.Furthermore, Z(x, θ * ) possesses one increasing sequence of poles starting at x = −θ 0 − θ * − θ t which lies in the half-plane Im x ≥ 0. The real parts of the poles in this sequence tend to +∞ as θ * → −∞.
Let K σs and K ρ be compact subsets of C\∆ S,σs and {Im ρ > −Q/2}\∆ ρ , respectively.Suppose (σ s , ρ) ∈ K σs × K ρ .Then, the above discussion shows that it is possible to choose a contour S = S(σ s , ρ) from −∞ to +∞ which separates the two upward from the three downward sequences of poles of Z(•, θ * )I S (•, σ s , ρ).Let us choose S so that its right tail approaches the horizontal line Im x = −Q/2 − δ as Re x → +∞ for some δ > 0. Then there is an N > 0 such that S is independent of θ * for θ * < −N , and (4.5) and (5.8) imply that, for all (σ s , ρ) Utilizing the asymptotic formula (2.8), it can be verified that the following limit holds uniformly for (σ s , ρ) ∈ K σs × K ρ and for x in bounded subsets of S.Moreover, using (2.8) with ǫ = 1/2 we find that I S obeys the estimates uniformly for (σ s , ρ) in compact subsets of K σs × K ρ and Im x in compact subsets of R. Since the contour S stays a bounded distance away from the poles of the integrand I S , we infer that there exists a constant uniformly for (σ s , ρ) in compact subsets of K σs × K ρ .In particular, the integrand I S has exponential decay along the left and right tails of the contour S.
Suppose we can show that there exist constants c > 0 and C > 0 such that uniformly for all θ * < −N , x ∈ S and (σ s , ρ) ∈ K σs × K ρ .Then it follows from (5.10), (5.11) and Lebesgue's dominated convergence theorem that the limit in (5.3) exists uniformly for (σ s , ρ) ∈ K σs × K ρ and is given by (5.4).This proves that the limit in (5.3) exists uniformly for (σ s , ρ) in compact subsets of D S and proves (5.4) for (σ s , ρ) ∈ D S .The analyticity of S as a function of (σ s , ρ) ∈ D S follows from the analyticity of H together with the uniform convergence on compact subsets.Whenever the condition (5.7) holds, the estimate (5.13) ensures that the integral representation (5.4) provides a meromorphic continuation of S to (σ s , ρ) ∈ C 2 which is analytic for (σ s , ρ) ∈ (C\∆ S,σs ) × (C\∆ ρ ).
(5.29b) 5.3.Polynomial limits.Our next two theorems state that S reduces to the Al-Salam Chihara polynomials when the variable ρ is suitably discretized and to the little q-Jacobi polynomials when σ 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 S in (5.6) are simple.
Theorem 5.7 (From S to the little q-Jacobi polynomials).Let ρ ∈ C\∆ ρ and suppose that Assumptions 1.1 and 5.5 are satisfied.Under the parameter correspondence the function S satisfies, for each n ≥ 0, lim where σ (n) s is defined in (3.11) and where Y n are the little q-Jacobi polynomials defined in (B.24).

The function X
In this section, we define the function X (b, θ, σ s , ω) 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 S. We show that 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 ω is suitably discretized.6.1.Definition and integral representation.Let θ and ω be two new parameters defined by Definition 6.1.The function X is defined by where S is given in (5.4).
The next theorem shows that for each choice of (b, θ) ∈ (0, ∞) × R, X is a well-defined analytic function of (σ s , ω) ∈ (C\∆ X ,σs ) × (C\∆ ω ), where ∆ S,σs , ∆ ω ⊂ C are discrete sets of points where 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 (σ s , ω) in compact subsets of where Moreover, the function X admits the following integral representation: where s b (ǫσ s − θ), (6.5) and the contour X is any curve from −∞ to +∞ which separates the three decreasing from the increasing sequences of poles, with the requirement that its right tail satisfies for some δ > 0. In particular, X is a meromorphic function of (σ s , ω) ∈ C 2 .If (σ s , ω) ∈ R 2 , then the contour X can be any curve from −∞ to +∞ lying within the strip Furthermore, as a consequence of (2.7), X satisfies X (b, θ, σ s , ω) = X (b −1 , θ, σ s , ω). (6.8) 6.2.Difference equations.The function X (b, θ, σ s , ω) is a joint eigenfunction of four difference operators, two acting on σ s and the remaining two on ω.This follows by taking the confluent limit (6.2) of the difference equations (5.25) and (5.29) satisfied by S. The proofs of the following two propositions are omitted since they are similar to those presented in Section 4.2.
6.2.1.First pair of equations.Introduce a difference operator H X (b, σ s ) such that where Proposition 6.3.For (σ s , ω) ∈ C 2 , the function X satisfies the following pair of difference equations: Second pair of equations.Introduce the difference operator HX (b, ω) such that HX (b, ω) = H+ X (b, ω)e ib∂ω + H− X (b, ω)e −ib∂ω + H0 X (b, ω), ( where Proposition 6.4.For (σ s , ω) ∈ C 2 , the function X satisfies the following pair of difference equations: (6.14b) 6.3.Polynomial limit.Our next theorem shows that X reduces to the continuous big q-Hermite polynomials when ω is suitably discretized.We make the following assumption, which implies that all the poles of the integrand I X are simple and that q = e 2iπb 2 is not a root of unity.
Assumption 6.5.Assume that b > 0 is such that b 2 is irrational and that The proof of the next theorem is analogous to the proof of Theorem 4.7 and is omitted.

The function Q
The function Q(b, σ s , η) is defined as a confluent limit of X and is one of the two elements at the fifth and lowest level of the non-polynomial scheme.We show that Q is a joint eigenfunction of four difference operators, two acting on σ s and two acting on η.Finally, we show that Q reduces to the continuous q-Hermite polynomials, which lie at the lowest level of the q-Askey scheme, when η is suitably discretized.
Interestingly, the mechanism behind the polynomial limit for 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 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 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 Q obtained by taking the confluent limit of the difference equations for X are "squares" of the simplest possible difference equations for 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).Moreover, define the normalization factor K by The factor K is a non-polynomial analog of the factor α −n in (B.40).
Definition 7.1.The function Q is defined by The next theorem shows that Q is a well-defined and analytic function of (σ s , η) ∈ C×({Im η < Q/2}\∆ η ), where ∆ η is a discrete set of points at which Q may have poles.The theorem also provides an integral representation for Q.Even if the requirement Im η < Q/2 is needed to ensure that the integral in the representation for Q converges, it will follow from the difference equations established later that Q extends to a meromorphic function of η ∈ C. Theorem 7.2.Suppose that Assumption 1.1 is satisfied.The limit (7.3) exists uniformly for (σ s , η) in compact subsets of where Moreover, the function Q admits the following integral representation: where and the contour of integration Q is any curve from −∞ to +∞ passing above the points x = ±σ s − iQ/2 and with the requirement that its right tail satisfies for some δ > 0. If (σ s , η) ∈ R 2 , then the contour Q can be any curve from −∞ to +∞ lying within the strip 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 θ → −∞ of the integral representation (6.4) for X , one should make the change of variables x → x − θ and thus write where Thanks to (2.7), we have (7.12) Remark 7.3.A close relative of the function Q has appeared in [16,42] in the context of quantum relativistic Toda systems.More precisely, the function H(a − , a + , x, y) in [42,Eq. (5.56)] is related to Q by where g b (z) satisfies s b (z) = g b (z)/g b (−z) and is defined by 7.2.Difference equations.We show that Q(b, σ s , η) is a joint eigenfunction of four difference operators, two acting on σ s and the remaining two on η.
We know from Theorem 7.2 that Q is a well-defined analytic function of σ s ∈ C and a meromorphic function of η for Im η < Q/2.The difference equations will first be derived as equalities between meromorphic functions defined on this limited domain.However, the difference equations in η can then be used to extend the results to η ∈ C, see Proposition 7.8.

First pair of equations. Define the difference operator H
where ) Proposition 7.4.For σ s ∈ C and Im η < Q/2, the function Q satisfies the following pair of difference equations: Proof.It is straightforward to verify that the following limits hold: where H + X and H 0 X are defined in (6.10) and (6.11), respectively.It follows that lim where H 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).
In what follows, we show that the function Q satisfies a pair of difference equations which is more fundamental than (7.18).Proposition 7.5.Define the difference operator ĤQ (b, σ s ) by with

.22)
The following operator identity holds: where H Q is defined in (7.15).
Proof.The left-hand side of (7.23) can be written as and straightforward computations show that the right-hand side coincides with the operator H Q (b, σ s ).
We next show that the difference equations (7.18) satisfied by the function Q can be simplified using the identity (7.23).The simplified equations can be viewed as "square roots" of the equations in (7.18).Proposition 7.6.For σ s ∈ C and Im η < Q/2, the function Q satisfies Proof.Let us rewrite (7.6) as where and where the contour Q is such that it passes above the points x = ±σ s , and such that Im x < Im η 2 −Q/4−δ for some small but fixed δ > 0 as Re x → +∞.The following estimates, which are easily established with the help of (2.8), imply that the integrand has exponential decay on Q as Re x → +∞: uniformly for (b, Im x, σ s , η) in compact subsets of (0, ∞) × R × C 2 .Utilizing the difference equation (2.5), we verify that the following identity holds: Letting the difference operator ĤQ act inside the contour integral in (7.26) and using (7.29), we obtain Performing the change of variables x = y + ib/2, we find Deforming the contour back to 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 Q.
The next proposition shows that Q extends to a meromorphic function of η 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 ∈ (0, ∞) and σ s ∈ C. Then there is a discrete subset ∆ ⊂ C such that the limit in (7.3) exists for all η ∈ C\∆.Moreover, the function Q defined by (7.3) is a meromorphic function of (σ s , η) ∈ C 2 and the four difference equations (7.25) and (7.31) hold as equalities between meromorphic functions of (σ s , η) ∈ C 2 .7.3.Polynomial limit.In this subsection, we show that the function Q reduces to the continuous q-Hermite polynomials when η is suitably discretized.Theorem 7.9 (From Q to the continuous q-Hermite polynomials).
For each integer n ≥ 0, the function Q satisfies 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.
There is a neighborhood U 0 of η 0 = iQ/2 and constants c > 0 and M > 0 for Re x ≤ 0 and for η ∈ U 0 with Im η ≤ Im η 0 , (7.39) uniformly for (Im x, σ s ) in compact subsets of R × C.
Proof of Theorem 7.9.Suppose b > 0 and σ s ∈ C. The function I Q (x, σ s , η) has two downward sequences of poles starting at x = ±σ s − iQ/2.Consider the representation (7.6) for Q with the contour Q passing above the points x = ±σ s − iQ/2 and satisfying (7.9) on its right tail.Taking ǫ = 1/2 in the estimate (7.36a), we find that there exists a constant M 1 such that uniformly for (b, σ s , η) in compact subsets of (0, ∞) × C 2 .Using (7.9) and noting that 2 ) e −2π(Im x)(Re η) e 2π(Im σs)(Re σs) , ( we infer the existence of a neighborhood U 0 of η 0 = iQ/2 and constants c > 0 and M > 0 such that for all x ∈ Q with Re x ≥ 0 and for all η ∈ U 0 . Writing , we can express (7.6) as The prefactor P Q defined in (7.7) has a simple zero at η = η 0 = iQ/2.Hence, by (7.44), the first term on the right-hand side of (7.45) vanishes in the limit η → η 0 .On the other hand, by (7.38), In view of Lemma 7.11, there exist constants M 3 , M 4 > 0 such that for all η in a small neighborhood of η 0 with Im η ≤ Im η 0 .By Proposition 7.8, Q is a meromorphic function of η ∈ C, and so has at most a pole at η 0 .To prove (7.35), it is therefore enough to consider the limit η → η 0 with η such that Im η < Im η 0 = Q/2.In the remainder of the proof, we assume Im η < Q/2.Then, after multiplication by the prefactor P Q , the second term on the right-hand side of (7.46) vanishes in the limit η → η 0 as a consequence of (7.47).Furthermore, employing (7.36b) and using that Im η < Im η 0 so that the contribution from −∞ + ia vanishes, we obtain The right-hand side of (7.48) has a simple pole at η = η 0 .Therefore, collecting the above conclusions, where the limits are taken with Im η < Im η 0 .Utilizing (7.7) and the identity Setting z = −η, recalling that η 0 = iQ/2, and using (2.4), we find which proves (7.35) for n = 0.
To show (7.35) also for n ≥ 1, we rewrite the difference equation (7.31a) as follows: Note that 1 + e 2πb(η− ib 2 ) vanishes for η = η 0 .Moreover, the function Q(b, σ s , η − ib) is analytic at η = η 0 .Indeed, as η approaches η 0 , the contour 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 Q retains its exponential decay provided that the right tail of the contour is deformed downwards.Therefore, evaluating (7.52) at η = η 0 and using (7.51), we obtain Q(b, σ s , η 1 ) = 2 cosh (2πbσ s ).Evaluating the recurrence relation (B.41) satisfied by the continuous q-Hermite polynomials at n = 0 and z = e 2πbσs , we find which proves (7.35) for n = 1.More generally, suppose that the function Q(b, σ s , η n ) exists for all n ≤ N and coincides with the polynomials Q n e 2πbσs ; q .Evaluating (7.52) at n = N + 1, we obtain and hence the recurrence relation (B.41) implies that By induction, we conclude that Q(b, σ s , η n ) exists for all n ≥ 0 and coincides with Q n e 2πbσs ; q .This completes the proof of the theorem.

The function L
In this section, we define the function L(b, θ t , θ, λ, µ) which generalizes the big q-Laguerre polynomials.It is defined as a confluent limit of H and lies at the third level of the non-polynomial scheme.We show that L is a joint eigenfunction of four difference operators.Finally, we show that L reduces to the big q-Laguerre polynomials, which lie at the third level of the q-Askey scheme, when λ is suitably discretized.8.1.Definition and integral representation.Let θ, Λ, λ, µ be defined as follows: Define the open set D L ⊂ C 2 by where the discrete subsets ∆ λ and ∆ µ are given by and is extended meromorphically to (λ, µ) ∈ C 2 .
The next theorem shows that L is a well-defined meromorphic function of (λ, µ) ∈ C 2 .
Theorem 8.2.Suppose that Assumption 1.1 is satisfied.The limit in (8.3) exists uniformly for (λ, µ) in compact subsets of D L .Moreover, L is an analytic function of (λ, µ) ∈ (C\∆ λ ) × (C\∆ µ ) and admits the following integral representation: where and the contour L is any curve from −∞ to +∞ which separates the three increasing from the two decreasing sequences of poles, with the requirement that its right tail satisfies for some δ > 0. In particular, L is a meromorphic function of (λ, µ) ∈ C 2 .If (λ, µ) ∈ R 2 , the contour L can be any curve from −∞ to +∞ lying within the strip Im x ∈ (−Q/2 + δ, 0).
Furthermore, the function L obeys the symmetry where and Proposition 8.3.For (λ, µ) ∈ C 2 , the function L satisfies the other pair of difference equations: 8.2.2.Second pair of difference equations.Define a difference operator HL (b, µ) such that where (8.18b) 8.3.Polynomial limit.Our next theorem, whose proof is similar to that of Theorem 4.7, shows that the function L reduces to the big q-Laguerre polynomials L n when λ 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 where L n are the big q-Laguerre polynomials defined in (B.29).

The function W
In this section, we define the function W(b, θ t , κ, ω) which generalizes the little q-Laguerre polynomials.It is defined as a confluent limit of L and lies at the fourth level of the non-polynomial scheme.We show that 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 κ is suitably discretized.
where 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, θ t ) ∈ (0, ∞) × R, W is a well-defined and analytic function of (κ, ω) ∈ (C\∆ κ ) × {Im ω < Q/2}, where ∆ κ ⊂ C is a discrete set of points at which W may have poles.In particular, W a meromorphic function of κ ∈ C and of ω for Im ω < Q/2.The theorem also provides an integral representation for W for (κ, ω) ∈ (C\∆ κ ) × {Im ω < Q/2}.In fact, even if the requirement Im ω < Q/2 is needed to ensure that the integral representation converges, it will be shown later, with the help of the difference equations, that W extends to a meromorphic function of (κ, ω) ∈ C 2 .Theorem 9.2.Suppose that Assumption 1.1 holds.The limit (9.2) exists uniformly for (κ, ω) in compact subsets of where Moreover, W is an analytic function of (κ, ω) ∈ D W and admits the following integral representation: where and the contour W is any curve from −∞ to +∞ which separates the decreasing sequence of poles from the two increasing ones, with the requirement that its right tail satisfies for some δ > 0. If (κ, ω) ∈ R 2 , then W can be any curve from −∞ to +∞ lying within the strip Im x ∈ (−Q/2 + δ, 0).
Furthermore, thanks to the symmetry (2.7), W satisfies (9.9) 9.2.Difference equations.By taking the confluent limit (9.2) of the difference equations (8.13) and (8.18) satisfied by L, we obtain the next two propositions which show that W is a joint eigenfunction of four difference operators, two acting on κ and the other two on ω.
The difference equations will first be derived as equalities between meromorphic functions of κ ∈ C and ω with Im ω < Q/2.We will then use the difference equations in ω to show that (i) the limit in (9.2) exists for all ω in the complex plane away from a discrete subset, (ii) W is in fact a meromorphic function of (κ, ω) in all of C 2 , and (iii) the four difference equations hold as equalities between meromorphic functions on C 2 , see Proposition 9.5.9.2.1.First pair of difference equations.Define the difference operator H W (b, κ) such that where (9.12) Proposition 9.3.For κ ∈ C and Im ω < Q/2, the function W satisfies the following pair of difference equations: The next proposition is stated without proof since it is similar to that of Proposition 4.5.
Proposition 9.5.Let (b, θ t ) ∈ (0, ∞) × R and κ ∈ C\∆ κ .There is a discrete subset ∆ ⊂ C such that the limit in (9.2) exists for all ω ∈ C\∆.Moreover, the function W defined by (9.2) is a meromorphic function of (κ, ω) ∈ C 2 and the four difference equations (9.13) and (9.18) hold as equalities between meromorphic functions of (κ, ω) ∈ C 2 .9.3.Polynomial limit.Our next theorem shows that W reduces to the little q-Laguerre polynomials when κ 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 Under the parameter correspondence the function W satisfies, for each n ≥ 0, where W n are the little q-Laguerre polynomials defined in (B.33).

The function M
In this section, we define the function M(b, ζ, ω) which generalizes the little q-Laguerre polynomials (B.33) evaluated at α = 0.It lies at the fifth and lowest level of the non-polynomial scheme and is defined as a limit of the function W. We show that M is a joint eigenfunction of four difference operators, two acting on ζ and the other two on ω.Finally, we show that M reduces to the little q-Laguerre polynomials evaluated at α = 0 when ζ is suitably discretized.
where the function W is defined in (9.2).
The next theorem shows that, for each choice of b ∈ (0, ∞), M is a well-defined and analytic function of where ∆ ζ ⊂ C is a discrete set of points at which M may have poles.More precisely, ∆ ζ is defined by The theorem also provides an integral representation for M for (ζ, ω) 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 M, that M extends to a meromorphic function of (ζ, ω) ∈ C 2 .
Then the limit (10.1) exists uniformly for (ζ, ω) in compact subsets of Moreover, M is an analytic function of (ζ, ω) ∈ D M and admits the following integral representation: where and the contour M is any curve from −∞ to +∞ which separates the increasing from the decreasing sequence of poles.If Im ω ∈ (0, Q/2) and ζ ∈ R, then the contour M can be any curve from −∞ to +∞ lying within the strip Im x ∈ (−Q/2, 0).
Thanks to the identity (2.7), M satisfies We know from Theorem 10.2 that M is a well-defined holomorphic function of ω for Im ω < Q/2 and is meromorphic in ζ for Im (ζ + ω) > 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 C 2 , see Proposition 10.5.

.10b)
Proof.The proof consists of taking the confluent limit (10.1) of the difference equation (9.13a).It is straightforward to verify that where H ± W is defined in (9.11).From (9.12), this implies Therefore we obtain where H M is given in (10.9).By Theorem 10.2, the limit in (10.Proof.The proof utilizes the difference equations in ζ and ω and is similar to the proof of Proposition 4.5.10.3.Polynomial limit.We now show that the function M reduces to the little q-Laguerre polynomials with α = 0 when ζ is suitably discretized.
Theorem 10.6 (From M to the little q-Laguerre polynomials with α .17) For each n ≥ 0, the function M satisfies where W n are the little q-Laguerre polynomials defined in (B.33) and where x Wn , q are given in (9.21).
Proof.We prove (10.18) by computing the limit ζ → ζ n of the representation (10.5) for each n ≥ 0. Let m, l ≥ 0 be integers and define x m,l ∈ C by The function s b (x − ζ) in (10.7) has a simple pole located at x = x m,l for any integers m, l ≥ 0. In the limit ζ → ζ n , the pole x n,0 collides with the pole of s b (x + iQ 2 ) located at x = 0, and the contour M is squeezed between the colliding poles.Hence, before taking the limit ζ → ζ n , we deform M into a contour M ′ which passes below x n,0 , thus picking up residue contributions from all the poles x = x m,l which satisfy Im x m,l ≥ Im x n,0 , i.e., from all the poles x m,l such that mb + l b ≤ nb.This leads to  e πb(2(m−n)ω−ibn 2 −imQ) q 1+n−m ; q m (q −m ; q) m .(10.24) We now apply the q-Pochhammer identity (α; q) n = (−α) n q n(n−1) 2 α −1 q 1−n ; q n (10.25) with α = q 1+n−m and with α = q −m to find e πb(2m−n)(2ω+ibn) e iπm(b 2 +2ibω−1) q −n ; q m (q; q) m .(10.26) After replacing m → n − m, in the sum and using the parameter correspondence (9.21), we obtain Using the q-Pochhammer identity with α = q −n and β = q, we arrive at Finally, utilizing the q-Pochhammer identity (10.25) with α = q −n , we find e iπn(b 2 (n+1)−1) (q −n ; q) n (q; q) n = 1.(10.30) Therefore we conclude that .31)This concludes the proof of (10.18).
12. Perspectives 12.1.Relations to other areas.We discuss some relations that deserve to be studied in more detail in the future.
12.1.1.Relation to quantum relativistic integrable systems.The function R was introduced in [37] in the context of relativistic systems of Calogero-Moser type, and studied in greater detail in [39][40][41].In particular, after a proper change of parameters, the difference operator H R defined in (3.6) corresponds to the rank one hyperbolic van Diejen Hamiltonian.Strong coupling (or Toda) limits of H R and of its higher rank generalizations were considered in [8].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 H 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 Q (see (7.6)), which is one of the two elements at the fifth level of the non-polynomial scheme, was studied in [16,42] and was interpreted as the eigenfunction of a q-Toda type Hamiltonian.

Figure 1 .
Figure1.The part of the q-Askey scheme relevant for the present paper.

. 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.[27, Theorem 4.2] Suppose Assumptions 1.1 and 3.1 are satisfied.Define {σ

5. 1 .
Definition and integral representation.Let ρ be a new parameter defined in terms of θ * and ν by ν = θ * 2 + ρ. (5.1) Define the open set D S ⊂ C 2 by

7. 1 .
Definition and integral representation.Let η be a new parameter defined as follows:

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 R defined in (3.1).Indeed, suppose Assumptions 1.1 and 3.1 are satisfied and define {σ

− 1 ,
(B.31)    withl + n = 1 − αq n+1 1 − βq n+1 , l − n = −αβq n+1 (1 − q n ) .(B.32) H,σs , ∆ ν ⊂ C are discrete sets of points at which H may have poles.In particular, H is a meromorphic function of σ s ∈ C and of ν for Im ν > −Q/2.More precisely, ∆ H,σs and ∆ ν are given by .8) 8.2.Difference equations.The next two propositions, whose proofs are omitted because they are similar to those presented in Section 4.2, show that the two pairs of difference equations (4.26) and (4.33) satisfied by the function H survive in the confluent limit (8.3), implying that L(b, θ t , θ, λ, µ) is a joint eigenfunction of four difference operators, two acting on λ and the other two on µ.The four difference equations hold as equalities between meromorphic functions of (λ, µ) ∈ C 2 .
8.2.1.First pair of difference equations.Define a difference operator H L (b, λ) such that .19) Theorem 8.6 (From L to the big q-Laguerre polynomials).Let µ ∈ {Im µ > −Q/2}\∆ µ and suppose that Assumptions 1.1 and 8.5 are satisfied.Define {λ n } ∞ n=0 ⊂ C by .19) Assumption 9.6 implies that all the poles of the integrand I W are distinct and simple.Theorem 9.7 (From W to the little q-Laguerre polynomials).Let ω ∈ C be such that Im ω < Q/2 and suppose that Assumptions 1.1 and 9.6 are satisfied.Define {κ n } ∞ n=0 ⊂ C by κ n = θ t + iQ 2 + ibn.(9.20) .8) 10.2.Difference equations.The two pairs of difference equations (9.13) and (9.18) satisfied by the function W survive in the confluent limit(10.1).This implies that M is a joint eigenfunction of four difference operators, two acting on ζ and two acting on ω.