1 Introduction

Postquantum calculus, or \((p,q)\)-calculus, is known as an extension of quantum calculus that recovers the results as \(p\to1\). For some basic properties of \((p,q)\)-calculus, we refer to [1–3].

In the q-case, the solutions of a q-Sturm-Liouville problem are q-orthogonal functions [4, 5], which reduce to the q-classical orthogonal polynomials, appear in a natural way [6]. Very recently [7], a new generalization of q-Sturm-Liouville problems, namely, \((p,q)\)-Sturm-Liouville problems, has been analyzed. In this paper, we show that the \((p,q)\)-difference equation is of hypergeometric type, that is, the \((p,q)\)-difference of any solution of the equation is also a solution of an equation of the same type. From this fundamental property the Rodrigues formula for the solutions is derived, and the coefficients of the three-term recurrence relation, satisfied by the orthogonal polynomial solutions of the \((p,q)\)-difference equation, are obtained.

The paper is organized as follows: In Section 2, we collect some definitions and notations of \((p,q)\)-calculus and include some new results that will be used in this paper. In Section 3, the \((p,q)\)-difference equations of hypergeometric type are introduced, in the sense that the \((p,q)\)-difference of a solution of the equation is solution of an equation of the same type. In Section 4, a Rodrigues-type formula for the polynomial solutions of the \((p,q)\)-difference equation of hypergeometric type is obtained. In Section 5, we obtain the coefficients in the three-term recurrence relation for the orthogonal polynomial solutions of the \((p,q)\)-difference equation of hypergeometric type. A difference representation and a \((p,q)\)-structure relation are also obtained. Finally, in Section 6, we present \((p,q)\)-analogues of shifted Jacobi, Laguerre, and Hermite polynomials. For each of this specific families, we provide a \((p,q)\)-difference equation of hypergeometric type, the coefficients of the three-term recurrence relation, the weight function, and the orthogonality property. Limit transitions from these \((p,q)\)-analogues to the classical families are also given. Appell families are also studied in detail.

2 Basic definitions and notations

In this section, we summarize the basic definitions and results, which can be found in [6, 8–12] and references therein.

For \(k \geq0\), the q-shifted factorial is defined as

$$ (a; q)_{k}=\prod_{j=0}^{k-1} \bigl(1-aq^{j}\bigr) \quad \text{with } (a; q)_{0}=1, $$
(1)

which can be generalized to the \((p,q)\)-power as

$$ \bigl((a,b);(p,q)\bigr)_{k}=\prod _{j=0}^{k-1} \bigl(ap^{j}-bq^{j} \bigr) \quad \text{with } \bigl((a,b);(p,q)\bigr)_{0}=1. $$
(2)

Moreover, for \(k<0\), we define

$$ \bigl((a,b);(p,q)\bigr)_{k}=\frac{1}{\prod_{j=0}^{-k} (ap^{-j}-bq^{-j})}. $$
(3)

Hence, we have

$$\begin{aligned}& \bigl((1,a);(1,q)\bigr)_{k}=(a; q)_{k}, \\& \bigl((ra,rb);(p,q)\bigr)_{k}=r^{k} \bigl((a,b);(p,q) \bigr)_{k}, \end{aligned}$$

and

$$(b/a;q/p)_{k}=a^{-k}p^{-k(k-1)/2}\bigl((a,b);(p,q) \bigr)_{k}. $$

Moreover,

$$(a; q)_{\infty} =\prod_{j=0}^{\infty}\bigl(1-aq^{j}\bigr)\quad \text{for } 0< |q|< 1 $$

can be generalized as

$$\bigl((a,b);(p,q)\bigr)_{\infty}=\prod_{j=0}^{\infty}\bigl(ap^{j}-bq^{j}\bigr) \quad \text{for } 0< \biggl\vert \frac{q}{p} \biggr\vert < 1. $$

For any complex number λ, we also introduce

$$ \bigl((a,b);(p,q)\bigr)_{\lambda}=\frac{((a,b);(p,q))_{\infty } }{((a p^{\lambda},b q^{\lambda});(p,q))_{\infty}}. $$
(4)

The q-numbers are defined as

$$\lim_{p\to1} [n]_{p,q}=[n]_{q}=\sum _{j=0}^{n-1} q^{j}, \quad q\neq1, $$

and their generalization as

$$ [n]_{p,q}=\frac{p^{n}-q^{n}}{p-q}=\sum _{j=0}^{n-1}q^{j}p^{n-1-j},\quad n=1,2, \ldots, $$
(5)

where

$$[-1]_{p,q}=-\frac{1}{pq} \quad \text{and}\quad [0]_{p,q}=0. $$

The \((p,q)\)-factorial is defined by

$$ [n]_{p,q}! = \prod_{j=1}^{n} [j]_{p,q},\quad n\geq1,\quad \text{and}\quad [0]_{p,q}!=1. $$
(6)

Since the definition of q-hypergeometric series

$$ {}_{r}\phi_{s}\left ( \textstyle\begin{array}{@{}c} {a_{1},\ldots,a_{r}} \\ {b_{1},\ldots,b_{s}} \end{array}\displaystyle \Bigm| {q};{z} \right ) = \sum_{j=0}^{\infty}\frac{(a_{1},\ldots,a_{r}; q)_{j}}{(b_{1},\ldots,b_{s}; q)_{j}}\frac{z^{j}}{(q;q)_{j}} \bigl((-1)^{j} q^{\frac{j(j-1)}{2}} \bigr)^{1+s-r}, $$

where

$$(a_{1},\ldots,a_{r};q)_{j}=(a_{1};q)_{j} \cdots(a_{r};q)_{j}, $$

is based on the symbol \((a;q)_{j}\) defined in (1), its generalization, known as the \((p,q)\)-hypergeometric series, can be defined as

$$\begin{aligned}& {}_{r}\Phi_{s}\left ( \textstyle\begin{array}{@{}c} {(a_{1p},a_{1q}),\ldots ,(a_{rp},a_{rq})}\\ {(b_{1p},b_{1q}),\ldots,(b_{sp},b_{sq})} \end{array}\displaystyle \Bigm| {(p,q)};{z} \right ) \\& \quad =\sum_{j=0}^{\infty}\frac{((a_{1p},a_{1q}),\ldots,(a_{rp},a_{rq});(p, q))_{j}}{((b_{1p},b_{1q}),\ldots,(b_{sp},b_{sq}); (p,q))_{j}} \frac {z^{j}}{((p,q);(p,q))_{j}} \bigl((-1)^{j} (q/p)^{\frac{j(j-1)}{2}} \bigr)^{1+s-r}, \end{aligned}$$
(7)

where

$$\bigl((a_{1p},a_{1q}),\ldots,(a_{rp},a_{rq});(p, q)\bigr)_{j}=\bigl((a_{1p},a_{1q});(p, q) \bigr)_{j}\cdots\bigl((a_{rp},a_{rq});(p, q) \bigr)_{j}, $$

and \(r, s \in\mathbb{Z}_{+}\) and \(a_{1p},a_{1q},\ldots ,a_{rp},a_{rq},b_{1p},b_{1q},\ldots,b_{sp},b_{sq},z \in\mathbb{C}\).

It is clear that

$$ \lim_{q \to1} {}_{r} \phi_{s}\left ( \textstyle\begin{array}{@{}c} {q^{a_{1}},\ldots,q^{a_{r}}}\\ {q^{b_{1}},\ldots ,q^{b_{s}}} \end{array}\displaystyle \Bigm| {q};{(q-1)^{1+s-r}z} \right ) = {}_{r}F_{s} \left ( \textstyle\begin{array}{@{}c} {{a_{1},\ldots,a_{r}}}\\ {{b_{1},\ldots,b_{s}}} \end{array}\displaystyle \Bigm| z \right ), $$
(8)

where

$$ {}_{r}F_{s}\left ( \textstyle\begin{array}{@{}c} {{a_{1},\ldots,a_{r}}}\\ {{b_{1},\ldots,b_{s}}} \end{array}\displaystyle \Bigm| z \right )=\sum_{j=0}^{\infty}\frac{(a_{1},\ldots,a_{r})_{j}}{(b_{1},\ldots,b_{s} )_{j}}\frac{z^{j}}{j!} $$

denotes a hypergeometric series with

$$(a_{1},\ldots,a_{r})_{j}=(a_{1})_{j} \cdots(a_{r})_{j}. $$

Also, when \(a_{1p}=a_{2p}=\cdots=a_{rp}=b_{1p}=b_{2p}=\cdots=b_{sp}=1\), \(a_{1q}=a_{1}, \ldots, a_{rq}=a_{r}\) and \(b_{1q}=b_{1},\ldots, b_{s,q}=b_{s}\), we have

$$ \lim_{p\to1} {}_{r}\Phi_{s}\left ( \textstyle\begin{array}{@{}c} {(1,a_{1}),\ldots ,(1,a_{r})}\\ {(1,b_{1}),\ldots,(1,b_{s})} \end{array}\displaystyle \Bigm| {(p,q)};{z} \right ) = {}_{r}\phi_{s}\left ( \textstyle\begin{array}{@{}c} {a_{1},\ldots,a_{r}}\\ {b_{1},\ldots,b_{s}} \end{array}\displaystyle \Bigm| {q};{z} \right ). $$

The functions

$$ E_{q}(x):= \sum_{n=0}^{\infty}\frac{q^{\frac{1}{2}n(n-1)}}{(q; q)_{n}}x^{n}=(-x;q)_{\infty}\quad \bigl( 0< \vert q \vert< 1 \text{ and } \vert x \vert< 1 \bigr) $$
(9)

and

$$ E_{p,q}(x):= \sum_{n=0}^{\infty}\frac{q^{\frac{1}{2}n(n-1)}}{((p,q);(p,q))_{n}}x^{n}=\bigl((1,-x);(p,q)\bigr)_{\infty}\quad \biggl( 0< \biggl\vert \frac {q}{p} \biggr\vert < 1 \text{ and } \vert x \vert< 1 \biggr) $$
(10)

are respectively known as a q-analogue and a \((p,q)\)-analogue of the exponential function.

The \((p,q)\)-difference operator is defined by (see e.g. [9, 13])

$$ ({\mathcal{D}}_{p,q}f) (x)=\frac{{\mathcal {L}}_{p} f(x)-{\mathcal{L}}_{q} f(x)}{(p-q)x},\quad x \neq0, $$
(11)

where

$$ {\mathcal{L}}_{a}h(x)=h(ax) $$
(12)

and \(({\mathcal{D}}_{p,q}f)(0)=f'(0)\), provided that f is differentiable at 0.

The \((p,q)\)-difference operator is a linear operator: for any constants a and b, we have

$$\bigl({\mathcal{D}}_{p,q}(af+bg)\bigr) (x)=a({\mathcal {D}}_{p,q}f) (x)+b({\mathcal{D}}_{p,q}g) (x). $$

Moreover, it can be proved that

$$\begin{aligned} \bigl({\mathcal{D}}_{p,q}(fg)\bigr) (x) =&f(px) ({\mathcal {D}}_{p,q}g) (x)+g(qx) ({\mathcal{D}}_{p,q}f) (x) \\ =& g(px) ({\mathcal{D}}_{p,q} f) (x)+f(qx) ({\mathcal{D}}_{p,q}g) (x). \end{aligned}$$
(13)

The \((p,q)\)-integral is defined by

$$ \int_{0}^{x} f(t)\,d_{p,q}t=(p-q)x \sum _{j=0}^{\infty}\frac {q^{j}}{p^{j+1}}f\biggl( \frac{q^{j}}{p^{j+1}}x\biggr). $$
(14)

For two nonnegative numbers a and b with \(a< b\), definition (14) yields

$$\int_{a}^{b} f(x)\,d_{p,q}x= \int_{0}^{b} f(x) \,d_{p,q}x - \int_{0}^{a} f(x)\,d_{p,q}x. $$

A regular Sturm-Liouville problem of continuous type is a boundary value problem of the form

$$ \frac{d}{dx} \biggl(r(x)\frac{dy_{n}(x)}{dx} \biggr)+ \lambda_{n} w(x) y_{n}(x)=0\quad \bigl(r(x)>0, w(x)>0\bigr), $$
(15)

which is defined on an open interval, say \((a,b)\), with boundary conditions

$$ \alpha_{1} y(a) + \beta_{1} y'(a)=0,\qquad \alpha_{2} y(b) + \beta_{2} y'(b)=0, $$
(16)

where \(\alpha_{1}\), \(\alpha_{2}\) and \(\beta_{1}\), \(\beta_{2}\) are constant numbers, and \(r(x)\), \(r'(x)\), and \(w(x)\) in (15) are assumed to be continuous for \(x\in[a,b]\). In this sense, if \(y_{n}\) and \(y_{m}\) are two eigenfunctions of equation (15), then according to Sturm-Liouville theory [14], they are orthogonal with respect to the weight function \(w(x)\) under the given condition (16), that is, we have

$$ \int_{a}^{b} w(x) y_{n}(x) y_{m}(x) \, dx=d_{n}^{2} \delta_{mn}, $$
(17)

where \(d_{n}^{2}=\int_{a}^{b} w(x)y_{n}^{2}(x)\, dx\) denotes the norm square of the functions \(y_{n}\), and \(\delta_{mn}\) stands for the Kronecker delta.

The following result has been proved in [7].

Theorem 2.1

Let \(\{y_{n}(x;p,q)\}\) be a sequence of functions satisfying the equation

$$ A (x) \bigl({\mathcal{D}}_{p,q}^{2} y_{n}\bigr) (x;p,q)+ B (x) ({\mathcal {D}}_{p,q} y_{n}) (px;p,q)+ \bigl( \lambda_{n,p,q}C(x)+D(x) \bigr) y_{n}(pqx;p,q)=0, $$
(18)

where \(A (x)\), \(B(x)\), \(C(x) \), and \(D (x)\) are known functions, and \(\lambda_{n,p,q} \) is a sequence of constants, then

$$ \int_{a}^{b} w^{*}(x;p,q) y_{n}(x;p,q) y_{m}(x;p,q)\, d_{p,q} x= \biggl( \int_{a}^{b} w^{*}(x;p,q) y^{2}_{n}(x;p,q) \, d_{p,q} x \biggr) \delta_{n,m}, $$

where

$$ w^{*}(x;p,q)=w(x;p,q) {\mathcal{L}}_{pq}^{-1}C(x)=w(x;p,q) C \biggl(\frac{1}{pq}x \biggr), $$
(19)

and \(w(x;p,q) \) is a solution of the \((p,q)\)-Pearson difference equation

$$ \bigl({\mathcal{D}}_{p,q}\bigl({\mathcal{L}}_{p} w {\mathcal{L}}_{q}^{-1}A\bigr)\bigr) (x;p,q) =B (x){ \mathcal {L}}_{pq}w(x;p,q) , $$
(20)

which is equivalent to

$$ \frac{w(p^{2}x;p,q)}{w(pqx;p,q)}=\frac{A(x)+(p-q)x B(x)}{A(pq^{-1}x)}. $$

Of course, the weight function defined in (19) must be be positive, and

$$w\bigl(q^{-1}x;p,q\bigr) A\bigl(p^{-1}q^{-2}x \bigr) $$

must vanish at \(x=a,b\).

Remark 2.1

Let \(\theta(x;p,q) \) be a known and predetermined function. The solution of the difference equation

$$ \frac{w(p^{2}x)}{w(pqx)}=\theta(x;p,q) $$
(21)

can be represented as [7]

$$w(x)=\prod_{k=0}^{\infty}\theta\biggl( \frac{q^{k}}{p^{k+2}}x;p,q\biggr). $$

3 \((p,q)\)-Difference equations of hypergeometric type

First, from the definition of shift operator (12) we can be verify that

$${\mathcal{D}}_{p,q}\bigl({\mathcal{L}}_{q}f(x)\bigr)=q { \mathcal{L}}_{q}\bigl({\mathcal{D}}_{p,q}f(x) \bigr). $$

Let us assume in (18) that \(A(x)\) and \(B(x)\) are polynomials of degree at most 2 and 1, respectively, \(D(x)=0\), and \(C(x)=1\). For our purposes, it is convenient to consider a particular case of (18) as

$$ \sigma(x) \bigl({\mathcal{D}}_{p,q}^{2} y \bigr) (x)+\tau(x){\mathcal {L}}_{p} \bigl(({\mathcal{D}}_{p,q} y) (x) \bigr) +\lambda{\mathcal{L}}_{pq} y(x)=0, $$
(22)

where

$$ \sigma(x)=ax^{2}+bx+c\quad \text{and}\quad \tau(x)=dx+e $$
(23)

with \(d \neq0\). Let \(y(x)\) be a solution of (22), and let

$$ v_{1}(x)=({\mathcal{D}}_{p,q})y(x). $$
(24)

We prove that \(v_{1}(x)\) is also a solution of an equation of the same type as (22).

With notation (24), we can rewrite (22) as

$$ \sigma(x) ({\mathcal{D}}_{p,q}v_{1}) (x)+ \tau(x){\mathcal{L}}_{p}(v_{1}) (x) +\lambda{ \mathcal{L}}_{pq}y(x)=0. $$
(25)

If the \((p,q)\)-difference operator \({\mathcal{D}}_{p,q}\) is applied to the latter equation, then it yields

$$ {\mathcal{D}}_{p,q}\bigl(\sigma(x) ({\mathcal {D}}_{p,q}v_{1}) (x)\bigr)+{\mathcal{D}}_{p,q} \bigl(\tau(x){\mathcal{L}}_{p}(v_{1}) (x) \bigr) +{ \mathcal{D}}_{p,q} \bigl(\lambda{\mathcal{L}}_{pq}y(x) \bigr)=0. $$
(26)

Also, since

$$\begin{aligned}& {\mathcal{D}}_{p,q}\bigl(\sigma(x) ({\mathcal {D}}_{p,q}v_{1}) (x)\bigr)={\mathcal{L}}_{p}\bigl({\mathcal {D}}_{p,q}v_{1}(x) \bigr) ({\mathcal{D}}_{p,q}\sigma ) (x)+{\mathcal{L}}_{q} \bigl(\sigma(x)\bigr) \bigl(D^{2}_{p,q}v_{1} (x) \bigr), \end{aligned}$$
(27)
$$\begin{aligned}& {\mathcal{D}}_{p,q}\bigl(\tau(x){\mathcal {L}}_{p}(v_{1}) (x) \bigr) ={\mathcal{L}}_{p}\tau(x) p {\mathcal{L}}_{p} \bigl({\mathcal{D}}_{p,q} v_{1}(x)\bigr)+{ \mathcal{L}}_{pq}\bigl(v_{1}(x)\bigr) \bigl({\mathcal {D}}_{p,q}\tau(x)\bigr), \end{aligned}$$
(28)

and

$$ {\mathcal{D}}_{p,q}\bigl(\lambda{\mathcal {L}}_{pq}y(x) \bigr)=\lambda p q {\mathcal{L}}_{pq}\bigl(v_{1}(x)\bigr), $$
(29)

we obtain

$$ \bigl({\mathcal{L}}_{q} \sigma(x) \bigr) \bigl({ \mathcal {D}}_{p,q}^{2} v_{1}\bigr) (x)+\tau _{1}(x){\mathcal{L}}_{p}\bigl(({\mathcal {D}}_{p,q}v_{1}) (x) \bigr) +\mu_{1} { \mathcal{L}}_{pq}v_{1}(x)=0, $$
(30)

where

$$ \tau_{1}(x)= p{\mathcal{L}}_{p} \bigl( \tau(x)\bigr)+ \bigl({\mathcal{D}}_{p,q}\sigma(x)\bigr). $$
(31)

Therefore, \(v_{1}(x)\) defined in (24) is solution of an equation of the same type as (22).

If the above procedure is similarly iterated, then we conclude that \(v_{n}(x)={\mathcal{D}}_{p,q}^{n} y(x)\) is also a solution of the equation

$$ \bigl( {\mathcal{L}}_{q}^{n} \sigma(x) \bigr) \bigl({\mathcal {D}}_{p,q}^{2} v_{n}\bigr) (x)+\tau_{n}(x){\mathcal{L}}_{p}\bigl(({ \mathcal{D}}_{p,q}v_{n}) (x) \bigr) +\mu_{n} { \mathcal{L}}_{pq}v_{n}(x)=0, $$
(32)

where

$$ \tau_{n}(x)= p{\mathcal{L}}_{p} \bigl( \tau_{n-1}(x)\bigr)+ \bigl({\mathcal{D}}_{p,q} \sigma_{n-1}(x)\bigr). $$
(33)

Hence, it is proved by induction that \(v_{n}(x)\) satisfies

$$ \sigma_{n}(x) \bigl({\mathcal{D}}_{p,q}^{2} v_{n}\bigr) (x)+\tau_{n}(x){\mathcal{L}}_{p} \bigl(({\mathcal{D}}_{p,q}v_{n}) (x) \bigr) +\mu _{n} {\mathcal{L}}_{pq}v_{n}(x)=0, $$
(34)

where

$$ \sigma_{n}(x)=\sigma\bigl(q^{n} x\bigr), \qquad { \mathcal{D}}_{p,q}\sigma _{n}(x)=q^{n}\bigl(b+a q^{n} (p+q)x\bigr) $$
(35)

and

$$ \tau_{n}(x)=e p^{n} + b [n]_{p,q} + \bigl(d p^{2n}+ a\bigl(p^{n}+q^{n}\bigr) [n]_{p,q} \bigr)x. $$
(36)

4 Rodrigues-type representation for the polynomial solutions of equation (22)

Theorem 4.1

The polynomial solutions of equation (22) satisfy the Rodrigues-type formula

$$ y_{n}(x)=K_{n} {\mathcal{L}}_{pq}^{-n} D_{p,q}^{n} \Biggl({\mathcal{L}}_{p}^{n} w(x) \prod_{k=1}^{n} {\mathcal{L}}_{p}^{n-k} {\mathcal {L}}_{q}^{k-2} \sigma(x) \Biggr), $$
(37)

where

$$ K_{n}=\frac{(-1)^{n} (D_{p,q}^{n} y_{n})(x)}{ (pq)^{(\frac{n^{2}+n-2}{2})}\prod_{k=0}^{n-1} \mu_{k}}\quad \textit{with } \mu_{0}=\lambda. $$

Proof

Let \(w(z)\) and \(w_{n}(z)\) satisfy the following \((p,q)\)-Pearson difference equations:

$$ D_{p,q} \bigl({\mathcal{L}}_{p} w(x) {\mathcal {L}}_{q^{-1}} \sigma(x) \bigr) =\tau(x) {\mathcal{L}}_{pq}w(x) $$

and

$$ D_{p,q} \bigl({\mathcal{L}}_{p} {{w}_{n}}(x){\mathcal {L}}_{q^{n-1}} \sigma(x) \bigr) =\tau_{n} (x){ \mathcal{L}}_{pq}w_{n}(x). $$

Multiplying (25) and (32) by \(w(z)\) and \(w_{n}(z)\), we can rewrite the equations in a self-adjoint form as

$$ D_{p,q} \bigl({\mathcal{L}}_{p} w(x) {\mathcal {L}}_{q^{-1}} \sigma(x) (D_{p,q}y) (x) \bigr) + \lambda_{n} {\mathcal{L}}_{pq}w(x) {\mathcal {L}}_{pq} y(x)=0 $$
(38)

and

$$ D_{p,q} \bigl({\mathcal{L}}_{p} w_{n}(x){\mathcal {L}}_{q^{n-1}} \sigma(x) (D_{p,q}v_{n}) (x) \bigr) + \mu_{n} {\mathcal{L}}_{pq}w_{n}(x) { \mathcal{L}}_{pq} v_{n}(x)=0. $$
(39)

On the other hand, since

$$ w_{n+1}(x)= {\mathcal{L}}_{p} w_{n}(x) {\mathcal {L}}_{q}^{n-1} \sigma(x) $$
(40)

and

$$ v_{n+1}(x)=D_{p,q}v_{n}(x), $$
(41)

using (40) and (41), we can write (39) as

$$ {\mathcal{L}}_{pq}w_{n}(x) {\mathcal {L}}_{pq}v_{n}(x)=- \frac{1}{\mu_{n}}D_{p,q}\bigl(w_{n+1}(x)v_{n+1}(x) \bigr). $$

If \(y(x)\) is a polynomial of degree n, that is, \(y=y_{n}(x)\), then

$$ v_{m}(x)=y_{n}^{(m)}(x) \quad \text{and} \quad v_{n}(x)=y_{n}^{(n)}(z)= \mathrm{const.}, $$

and for \(y_{n}^{(m)}(x)\), we obtain

$$ D_{p,q}^{m}\bigl({\mathcal{L}}_{pq} y_{n}(x) \bigr)=K'_{n} {\mathcal{L}}_{pq}^{-(n-m-1)} D_{p,q}^{n-m}\bigl( w_{n}(x) \bigr), $$

where

$$ K'_{n}=\frac{(-1)^{n-m} (D_{p,q}^{n} y_{n})(x)}{ (pq)^{(\frac {n^{2}+n+2m-2}{2})}\prod_{k=m}^{n-1} \mu_{k}}. $$

The result follows from this expression for \(m=0\). □

5 Three-term recurrence relation for the polynomial solutions of equation (22)

First, to calculate the corresponding eigenvalues \(\lambda_{n,p,q}\), since

$$D_{p,q} \bigl(x^{n}\bigr)=\frac{p^{n} x^{n} -q^{n} x^{n}}{(p-q)x}=[n]_{p,q} x^{n-1}, $$

by equating the coefficients of \(x^{n}\) we obtain

$$ \lambda_{n,p,q}=-\frac{[n]_{p,q}}{(pq)^{n}} \bigl( a [n-1]_{p,q}+d p^{n-1} \bigr). $$
(42)

Lemma 5.1

For each nonnegative integer n, the uniqueness of a monic polynomial solution of equation (22) is equivalent to the following conditions:

  1. (1)

    The equation in j

    $$\lambda_{j,p,q}=\lambda_{n,p,q} $$

    has \(j=n\) as a unique solution in N;

  2. (2)

    \(\lambda_{k,p,q} \neq0\) for \(k=0,1,\ldots,n-1\).

Proof

The result can be obtained following the same steps as in the continuous case. □

Let us define a linear operator as

$$ L_{n}\bigl[y(x)\bigr]:=\bigl(a x^{2}+b x+c \bigr)D_{p,q}^{2} y(x;p,q)+ (d x+e ) D_{p,q} y(px;p,q)+\lambda_{n} y(pqx;p,q), $$
(43)

where \(\lambda_{n}=\lambda_{n,p,q}\) is defined in (42).

Lemma 5.2

There exists a sequence \(\{ \beta_{n} \}_{n \in{\mathbf{N}}}\) such that the polynomial

$$ U_{n}(x)=L_{n+1} \bigl((x- \beta_{n}) P_{n}(x)\bigr) $$
(44)

has exactly degree \(n-1\) for each \(n \in{\mathbf{N}}\) and

$$ \beta_{n}=\varpi_{1,n} + \frac{p^{-n} q^{-n} [n+1]_{p,q} ({b } [n]_{p,q}+{e } p^{n} )}{\lambda_{n+1}-\lambda_{n}}. $$
(45)

Moreover, \(U_{n}(x)=\vartheta_{n} x^{n-1} + \cdots\) with

$$\begin{aligned} \vartheta_{n} =&\frac{1}{p^{2} q} \bigl( \bigl(-p^{1 + n} q^{n} \lambda_{n+1} (\beta_{n} \varpi_{1,n} - \varpi_{2,n}) - p^{n} q \bigl(d ( \beta_{n} \varpi_{1,n} - \varpi_{2,n}) [n-1]_{p,q} \\ &{} + e p (\beta_{n} - \varpi_{1,n}) [n]_{p,q} \bigr) + p^{2} q \bigl([n-1]_{p,q} \bigl(a (- \beta_{n} \varpi_{1,n} + \varpi_{2,n}) [n-2]_{p,q} \\ &{}+ b (-\beta_{n} + \varpi_{1,n}) [n]_{p,q} \bigr) + c [n]_{p,q} [n+1]_{p,q}\bigr)\bigr) \bigr) \end{aligned}$$
(46)

and \(P_{n}(x)=x^{n} + \varpi_{1,n} x^{n-1} + \cdots\) .

Proof

Let us expand the monic polynomial solution of equation (42):

$$ y_{n}(x;p,q)=P_{n}(x)=x^{n} + \varpi_{1,n} x^{n-1} + \varpi_{2,n} x^{n-2} + \cdots. $$
(47)

Since

$$ (x-\beta_{n}) P_{n}(x)=x^{n+1} + x^{n} (\varpi_{1,n}-\beta_{n}) + x^{n-1} ( \varpi_{2,n} - \beta_{n} \varpi_{1,n}) + \cdots , $$

we have

$$\begin{aligned}& L_{n+1} \bigl[(x-\beta_{n})P_{n}(x)\bigr] \\& \quad = \bigl(p^{1 + n} q^{1 + n} \lambda_{n+1} + \bigl(d p^{n} + a [n]_{p,q}\bigr) [n+1]_{p,q}\bigr) x^{n+1} \\& \qquad {}+ \frac{1}{p} \bigl( \bigl(p^{1 + n} q^{n} \lambda_{n+1} (-\beta_{n} + \varpi_{1,n}) - d p^{n} \beta_{n} [n]_{p,q} \\& \qquad {}+ d p^{n} \varpi_{1,n} [n]_{p,q} - a p \beta_{n} [n-1]_{p,q} [n]_{p,q} \\& \qquad {}+ a p \varpi_{1,n} [n-1]_{p,q} [n]_{p,q} + e p^{1 + n} [n+1]_{p,q} + b p [n]_{p,q} [n+1]_{p,q}\bigr) \bigr) x^{n} \\& \qquad {}+ \frac{1}{p^{2} q} \bigl( \bigl(-p^{1 + n} q^{n} \lambda_{n+1} (\beta_{n} \varpi_{1,n} - \varpi_{2,n}) - p^{n} q \bigl(d (\beta_{n} \varpi_{1,n} - \varpi_{2,n}) [n-1]_{p,q} \\& \qquad {} + e p (\beta_{n} - \varpi_{1,n}) [n]_{p,q}\bigr) + p^{2} q \bigl( [n-1]_{p,q} \bigl(a (-\beta_{n} \varpi_{1,n} + \varpi_{2,n}) [n-2]_{p,q} \\& \qquad {} + b (-\beta_{n} + \varpi_{1,n}) [n]_{p,q}\bigr) + c [n]_{p,q} [n+1]_{p,q}\bigr) \bigr)\bigr) x^{n-1}. \end{aligned}$$
(48)

The coefficient in \(x^{n+1}\) in (48) is zero by noting the value of \(\lambda_{n}\) in (42). To have a polynomial of degree exactly \(n-1\) in the variable x, we obtain (45) with the condition \(\lambda_{n+1} \neq\lambda_{n}\). Finally, the coefficient of \(x^{n-1}\) is derived by (46). □

Lemma 5.3

For each nonnegative integer n, we have

$$L_{n-1}\bigl(U_{n}(x)\bigr)=0, $$

where \(U_{n}(x)\) is defined in (44).

By the uniqueness of the polynomial solution of (22) there exists a constant \(\Omega_{n}\) such that

$$U_{n}(x)=\Omega_{n} P_{n-1}(x). $$

Lemma 5.4

Let \(\bar{P}_{n}(x)\) be the unique monic polynomial solution of degree n of (22). Then, there exist two sequences \(\{ \beta_{n} \} _{n \geq0}\) and \(\{ \gamma_{n} \}_{n \geq1}\) such that the following three-term recurrence relation holds:

$$ \bar{P}_{n+1}(x)=(x-\beta_{n}) \bar{P}_{n}(x)-\gamma_{n} \bar{P}_{n-1}(x). $$
(49)

Moreover, \(\beta_{n}\) is given in (45), and

$$ \gamma_{n}=\frac{\Omega_{n}}{\lambda_{n-1}-\lambda_{n+1}}. $$
(50)

These two lemmas can be improved as follows.

Theorem 5.1

Let \(\bar{P}_{n}(x)\) be the monic polynomial solution of degree n of (22), where \(\sigma(x)\) and \(\tau(x)\) are given in (23), and \(\lambda_{n}\) is given in (42). Then, the coefficients \(\beta_{n}\) and \(\gamma_{n}\) of the three-term recurrence relation (49) are explicitly given by

$$ \beta_{n}=\varpi_{1,n}-\varpi_{1,n+1} $$
(51)

and

$$ \gamma_{n}=\varpi_{2,n}-\varpi_{2,n+1}- \beta_{n} \varpi_{1,n}, $$
(52)

where

$$ \varpi_{1,n}=-\frac{p q [n]_{p,q} (b p [n-1]_{p,q}+e p^{n} )}{[n-1]_{p,q} (a p (p q [n-2]_{p,q}-[n]_{p,q})+d q p^{n} )-d p^{n} [n]_{p,q}} $$
(53)

and

$$ \varpi_{2,n}=-\frac{p q^{2} [n-1]_{p,q} (\varpi_{1,n} (b p^{2} [n-2]_{p,q}+e p^{n} )+c p^{2} [n]_{p,q} )}{q^{2} [n-2]_{p,q} (a p^{3} [n-3]_{p,q}+d p^{n} )-[n]_{p,q} (a p [n-1]_{p,q}+d p^{n} )}. $$
(54)

Next, we obtain the \((p,q)\)-difference representation for the polynomial solutions of (22).

Theorem 5.2

Let \(P_{n}(x)\) be the unique monic polynomial solution of (22). Then, the following relation holds:

$$ P_{n}(px)=U_{n} {\mathcal{D}}_{p,q}P_{n+1}(x) + V_{n} {\mathcal{D}}_{p,q}P_{n}(x) + W_{n} {\mathcal{D}}_{p,q} P_{n-1}(x),\quad n \geq2, $$
(55)

where

$$\begin{aligned}& U_{n} =\frac{p^{n}}{[n+1]_{p,q}}, \end{aligned}$$
(56)
$$\begin{aligned}& V_{n} =p^{n} \biggl( \frac{\varpi_{1,n}}{p [n]_{p,q}} - \frac {\varpi_{1,n+1}}{[n+1]_{p,q}} \biggr), \end{aligned}$$
(57)
$$\begin{aligned}& W_{n} =p^{n} \biggl(-\frac{\varpi_{1,n}^{2}}{p [n]_{p,q}} + \frac {\varpi_{2,n}}{p^{2} [n-1]_{p,q}} + \frac{\varpi _{1,n}\varpi _{1,n+1}-\varpi_{2,n+1}}{[n+1]_{p,q}} \biggr), \end{aligned}$$
(58)

and \(\varpi_{1,n}\) and \(\varpi_{2,n}\) are explicitly given in (53) and (54).

Proof

The result follows by equating the coefficients of (55). □

Moreover, the polynomial solutions of (22) also satisfy a \((p,q)\)-structure relation.

Theorem 5.3

Let \(P_{n}(x)\) be the unique monic polynomial solution of (22). Then, the following relation holds:

$$ \phi(x) {\mathcal{D}}_{p,q}P_{n} \biggl( \frac{x}{p} \biggr)=\hat{U}_{n} P_{n+1}(x) + \hat{V}_{n} P_{n}(x) + \hat{W}_{n} P_{n-1}(x),\quad n \geq1, $$
(59)

where

$$ \phi(x)=a x^{2}+b p q x+c p^{2}q^{2}, $$
(60)

and the coefficients are explicitly given by

$$\begin{aligned}& \hat{U}_{n} =a p^{1-n} [n]_{p,q}, \end{aligned}$$
(61)
$$\begin{aligned}& \hat{V}_{n} =p^{1-n} \bigl(a p [n-1]_{p,q} \varpi _{1,n}+[n]_{p,q}(b p q-a \varpi_{1,n+1}) \bigr), \end{aligned}$$
(62)
$$\begin{aligned}& \hat{W}_{n} =p^{1 - n} \bigl(p \bigl([n-1]_{p,q} \varpi_{1,n} (b p q - a \varpi_{1,n}) + a p [n-2]_{p,q} \varpi_{2,n}\bigr) \\& \hphantom{\hat{W}_{n} ={}}{}+ [n]_{p,q}\bigl(c p^{2} q^{2} + \varpi_{1,n} (-b p q + a \varpi _{1,n+1}) - a \varpi_{2,n+1}\bigr) \bigr), \end{aligned}$$
(63)

where \(\varpi_{1,n}\) and \(\varpi_{2,n}\) are given in (53) and (54), respectively.

Proof

The result follows by equating the coefficients of (59). □

6 Examples

6.1 Example 1: Appell families

If \(\{P_{n}(x)\}_{n \in{\mathbf{N}}}\) is a polynomial solution of (22) such that

$$ {\mathcal{D}}_{p,q}P_{n}(x)=[n]_{p,q} P_{n-1}(x), $$
(64)

then the solution of (64) is said to be of Appell type.

To find these families, by the \((p,q)\)-difference representation (55) the above condition (64) is equivalent to \(V_{n}=W_{n}=0\) for all n.

By equating \(V_{1}=0\), since \(p \neq0\) and \(q \neq0\), we obtain three following possibilities:

  1. (i)

    \(a=b=0\), which implies that \(V_{n}=W_{n}=0\). In this case, since \(d \neq0\), we can conclude that the coefficients of the three-term recurrence relation (49) are given by

    $$ \beta_{n}=-\frac{e p^{1-n} q^{n+1}}{d}\quad \text{and} \quad \gamma_{n}=-\frac {c p^{3-2 n} q^{n+1} }{d }[n]_{p,q}, $$
    (65)

    assuming that \(p \neq q\). Notice that

    $$\lim_{p \to q} \gamma_{n}=\lim_{p \to q} -\frac{c p^{3-2 n} q^{n+1} }{d }[n]_{p,q}=-\frac{c n q^{3}}{d}. $$
  2. (ii)

    \(b=e=0\), which implies that \(V_{n}=0\). In order that \(W_{n}=0\), we must analyze three cases,

    1. (a)

      \(a=0\), which implies

      $$\beta_{n}=0 \quad \text{and} \quad \gamma_{n}=- \frac{c p^{3-2 n} q^{n+1}}{d }[n]_{p,q}, $$

      assuming that \(p \neq q\);

    2. (b)

      \(c=0\), which implies \(\gamma_{n}=0\), and therefore we have no orthogonal polynomial sequences;

    3. (c)

      \(p \to q\), for which we also need \(c=0\) in order to have \(W_{n}=0\). Therefore we have no orthogonal polynomial sequences again.

  3. (iii)

    \({q=\frac{bdp-aep}{a e}}\), assuming that \(a \neq0\) and \(e \neq0\), which gives no orthogonal polynomial sequence after imposing that \(V_{n}=W_{n}=0\) for \(n \geq2\).

As a consequence of this analysis, we observe that the unique possibility for having \((p,q)\)-Appell families is \(a=b=0\), which contains as a particular case the symmetric option \(a=b=e=0\). It is possible to assume that \(c=1\) without loss of generality.

Theorem 6.1

The polynomial solution of equation (22) in the cases \(a=b=e=0\) and \(c=1\) is explicitly given by

$$ y_{n}(x;p,q)=x^{\sigma_{n}} {}_{2} \Phi_{1}\left ( \textstyle\begin{array}{@{}c}{(p^{\sigma_{n}-n},q^{\sigma _{n}-n}),(d p^{2 [{(n-1)}/{2} ]+1},0)} \\ {(p ^{2\sigma _{n}+1},q^{2\sigma _{n}+1})} \end{array}\displaystyle \Bigm| { \bigl(p^{2},q^{2}\bigr)};{(q-p)x^{2}} \right ), $$
(66)

up to a normalizing constant, where

$$\sigma_{n} =\frac{1-(-1)^{n}}{2}= \textstyle\begin{cases} 0, & n \textit{ even}, \\ 1, & n \textit{ odd}. \end{cases} $$

In this case, the Pearson-type \((p,q)\)-difference equation reads as

$$ \bigl({\mathcal{D}}_{p,q}({\mathcal{L}}_{p} w)\bigr) (x;p,q) =dx {\mathcal{L}}_{pq} w(x;p,q), $$

where

$$ w(x;p,q)=\sum_{n=0}^{\infty} \frac{d^{n} q^{n(n-1)}}{p^{2n} \prod_{j=1}^{n} [2j]_{p,q}} x^{2n} = E_{p^{2},q^{2}} \bigl((p-q)p^{-2}dx^{2} \bigr) $$
(67)

with \(E_{p,q}\) defined in (10).

Remark 6.1

We emphasize that as \((p,q)\to(1,1)\), for \(d=-2\), the second-order \((p,q)\)-difference equation

$$ \bigl({\mathcal{D}}_{p,q}^{2} y\bigr) (x)+dx {\mathcal{L}}_{p} \bigl(({\mathcal{D}}_{p,q}y) (x) \bigr) - \frac{d q^{-n}}{p} [n]_{p,q} {\mathcal{L}}_{pq} y(x)=0 $$
(68)

converges formally to the differential equation of Hermite polynomials. Moreover, the polynomials \(y_{n}(x;p,q)\) defined in (66) converge to the well-known Hermite polynomials, and the weight function \(w(x;p,q)\) defined in (67) converges to \(\exp(-x^{2})\).

The monic polynomial solutions of (68) satisfy a three-term recurrence relation of the form

$$ y_{n+1}(x;p,q)=x y_{n}(x;p,q) - C_{n}(p,q) y_{n-1}(x;p,q) $$

with

$$y_{0}(x;p,q)=1,\qquad y_{1}(x;p,q)=x, $$

where

$$ C_{n}(p,q)=-\frac{p^{3-2 n} q^{n+1}}{d} [n]_{p,q}. $$

To have the orthogonality with respect to a positive weight function, we need to impose \(d<0\). Under this assumption, the orthogonality reads as

$$ \int_{-\infty}^{\infty} y_{n}(x;p,q) y_{m}(x;p,q) E_{p^{2},q^{2}} \bigl((p-q)p^{-2} dx^{2}\bigr) \,d_{p,q}x=c_{0} \biggl( \frac{-1}{d} \biggr)^{n} \frac {q^{\frac{1}{2} n (n+3)}}{p^{(n-2) n}} [n]_{p,q}! \delta_{n,m}, $$

where

$$c_{0}= \int_{-\infty}^{\infty} E_{p^{2},q^{2}} \bigl((p-q)p^{-2} dx^{2}\bigr) \,d_{p,q}x, $$

and \([z]_{p,q}!\) is defined in (6).

6.2 Example 2: \((p,q)\)-Laguerre polynomials

Let us now consider the second-order equation

$$ x \bigl({\mathcal{D}}_{p,q}^{2} y\bigr) (x)+ \biggl( \frac{p^{\alpha+1} q^{-\alpha-1}-1}{p-q}+d x \biggr) {\mathcal{L}}_{p} \bigl(({ \mathcal{D}}_{p,q}y) (x) \bigr) -\frac{d q^{-n}}{p} [n]_{p,q} {\mathcal{L}}_{pq} y(x)=0. $$
(69)

Theorem 6.2

The polynomial solution of (69) is given by

$$ y_{n}(x;\alpha;p,q)= {}_{2} \Phi_{1}\left ( \textstyle\begin{array}{@{}c} {(p^{-n},q^{-n}),(p^{n-1},0)}\\ {(p ^{\alpha +1},q^{\alpha+1})} \end{array}\displaystyle \Bigm| {(p,q)};{d q^{\alpha+1} (q-p) x } \right ) $$
(70)

up to a normalizing constant.

In this case, the Pearson-type \((p,q)\)-difference equation reads as

$$ \frac{w(p^{2}x;\alpha;p,q)}{w(pqx;\alpha;p,q)}=p^{\alpha} q^{-\alpha}-\frac{d q^{2} x}{p}+d q x, $$

in which

$$ w(x;\alpha;p,q)=x^{\alpha} E_{p,q} \bigl(d x p^{-\alpha-3} q^{\alpha +1} (p-q) \bigr). $$
(71)

Remark 6.2

Once again, we emphasize that as \((p,q)\to(1,1)\), for \(d=1\), the second-order \((p,q)\)-difference equation (69) converges formally to the differential equation of Laguerre polynomials. Moreover, the polynomials \(y_{n}(x;\alpha;p,q)\) defined in (70) converge to the well-known Laguerre polynomials, and the weight function \(w(x;\alpha;p,q)\) defined in (71) converges to \(x^{\alpha} \exp(-x)\).

The monic polynomial solutions of equation (69) satisfy a three-term recurrence relation of the form

$$ y_{n+1}(x;\alpha;p,q)=\bigl(x-B_{n}(\alpha;p,q)\bigr) y_{n}(x;\alpha;p,q) - C_{n}(\alpha;p,q) y_{n-1}(x; \alpha;p,q) $$

with

$$y_{0}(x;\alpha;p,q)=1,\qquad y_{1}(x;\alpha;p,q)=x-B_{0}( \alpha;p,q), $$

where

$$ B_{n}(\alpha;p,q)=\frac{p^{1-2 n} q^{n} (q^{n} (p+q)-p^{n+1} (p^{\alpha} q^{-\alpha}+1 ) )}{d (p-q)} $$

and

$$ C_{n}(\alpha;p,q)=\frac{p^{5-4 n} q^{-\alpha+2 n-1} [n]_{p,q} [\alpha+n]_{p,q}}{d^{2}}. $$

To have orthogonality with respect to a positive weight function, we need to impose \(\alpha>-1\). Under this assumption, the orthogonality reads as

$$\begin{aligned}& \int_{0}^{\infty} y_{n}(x;\alpha;p,q) y_{m}(x;\alpha;p,q) x^{\alpha } E_{p,q} \bigl(d x p^{-\alpha-3} q^{\alpha+1} (p-q) \bigr) \,d_{p,q}x \\& \quad =c_{0}(\alpha) \frac{p^{(3-2 n) n} q^{n (n-\alpha )}}{d^{2}n} [n]_{p,q}! [n+ \alpha]_{p,q}! \delta_{n,m}, \end{aligned}$$

where

$$c_{0}(\alpha)= \int_{0}^{\infty} x^{\alpha} E_{p,q} \bigl(d x p^{-\alpha-3} q^{\alpha+1} (p-q) \bigr) \,d_{p,q}x. $$

6.3 Example 3: \((p,q)\)-shifted Jacobi polynomials

Consider the second-order \((p,q)\)-difference equation

$$\begin{aligned}& \frac{q x (q x-p)}{p^{2}} \bigl({\mathcal{D}}_{p,q}^{2} y\bigr) (x)+ \biggl( \frac {x p^{\alpha+\beta+2} q^{-\alpha-\beta}-p^{\beta+2} q^{-\beta}+p q-q^{2} x}{p^{2} (p-q)} \biggr) {\mathcal{L}}_{p} \bigl(({ \mathcal{D}}_{p,q}y) (x) \bigr) \\& \quad {}+ [n]_{p,q} \biggl(\frac{q p^{-n-2}-p^{\alpha+\beta-1} q^{-\alpha -\beta-n}}{p-q} \biggr) { \mathcal{L}}_{pq} y(x)=0. \end{aligned}$$
(72)

Theorem 6.3

The polynomial solution of (72) is given by

$$ y_{n}(x;\alpha,\beta;p,q)= {}_{2} \Phi_{1}\left ( \textstyle\begin{array}{@{}c} {(p^{-n},q^{-n}),(p ^{\alpha +\beta+n+1},q^{\alpha +\beta+n+1})}\\ {(p ^{\beta+1},q^{\beta +1})} \end{array}\displaystyle \Bigm| {(p,q)};{\frac{x q^{-\alpha}}{p} } \right ) $$
(73)

up to a normalizing constant.

In this case, the Pearson-type \((p,q)\)-difference equation reads as

$$ \frac{w(p^{2}x;\alpha;p,q)}{w(pqx;\alpha;p,q)}= \frac{p^{\beta} q^{-\alpha-\beta} (x p^{\alpha}-q^{\alpha} )}{x-1}, $$

where

$$ w(x;\alpha;p,q)=\frac{x^{\beta}}{((1,x p^{-2});(p,q))_{-\alpha}}, $$
(74)

and \(((a,b);(p,q))_{\lambda}\) is defined in (4).

Remark 6.3

It is straightforward to check that as \((p,q)\to(1,1)\), the second-order \((p,q)\)-difference equation (72) converges formally to the differential equation of shifted Jacobi polynomials. Moreover, the polynomials \(y_{n}(x;\alpha,\beta;p,q)\) defined in (73) converge to the well-known shifted Jacobi polynomials, and the weight function \(w(x;\alpha,\beta;p,q)\) defined in (74) converges to \(x^{\alpha} (1-x)^{\beta}\).

The monic polynomial solutions of equation (72) satisfy a three-term recurrence relation of the form

$$ y_{n+1}(x;\alpha,\beta;p,q)=\bigl(x-B_{n}(\alpha,\beta;p,q) \bigr) y_{n}(x;\alpha,\beta;p,q) - C_{n}(\alpha,\beta;p,q) y_{n-1}(x;\alpha,\beta;p,q) $$

with

$$ y_{0}(x;\alpha,\beta;p,q)=1,\qquad y_{1}(x;\alpha,\beta ;p,q)=x-B_{0}(\alpha,\beta;p,q), $$

where

$$\begin{aligned}& B_{n} (\alpha,\beta;p,q)=\frac{p^{n+2} q^{\alpha+n+1}}{(p-q)^{2} [\alpha+\beta+2 n]_{p,q} [\alpha+\beta+2 n+2]_{p,q}} \\& \hphantom{B_{n} (\alpha,\beta;p,q)={}}{}\times \bigl( \bigl(p^{\beta}+q^{\beta} \bigr) q^{\alpha+\beta +2 n+1}-(p+q) \bigl(p^{\alpha}+q^{\alpha} \bigr) p^{\beta+n} q^{\beta+n} \\& \hphantom{B_{n} (\alpha,\beta;p,q)={}}{}+ \bigl(p^{\beta}+q^{\beta} \bigr) p^{\alpha+\beta+2 n+1} \bigr), \\& C_{n}(\alpha,\beta;p,q)=\frac{p^{\beta+2 n+3} q^{2 \alpha+\beta+2 n+1} [n]_{p,q} [\alpha+n]_{p,q} [\beta +n]_{p,q} [\alpha +\beta+n]_{p,q}}{[\alpha+\beta +2 n-1]_{p,q} ([\alpha+\beta+2 n]_{p,q})^{2} [\alpha+\beta+2 n+1]_{p,q}}. \end{aligned}$$

To have the orthogonality with respect to a positive weight function, we need to impose \(\alpha,\beta>-1\). Under these assumptions, the orthogonality reads as

$$\begin{aligned}& \int_{0}^{p/q} y_{n}(x;\alpha,\beta;p,q) y_{m}(x;\alpha,\beta;p,q) \frac{x^{\beta}}{((1,x p^{-2});(p,q))_{-\alpha}} \,d_{p,q}x \\& \quad =c_{0}(\alpha,\beta) \frac{p^{n (\beta+n+4)} q^{n (2 \alpha+\beta +n+2)} [n]_{p,q}! [\alpha+n]_{p,q}! [\beta +n]_{p,q}! [\alpha+\beta+n]_{p,q}!}{[\alpha +\beta+2 n-1]_{p,q}! ([\alpha +\beta+2 n]_{p,q}!)^{2} [\alpha+\beta+2 n+1]_{p,q}!} \delta_{n,m}, \end{aligned}$$

where

$$c_{0}(\alpha,\beta)= \int_{0}^{p/q} \frac{x^{\beta}}{((1,x p^{-2});(p,q))_{-\alpha}} \,d_{p,q}x. $$