Open Access

On \((p,q)\)-classical orthogonal polynomials and their characterization theorems

Advances in Difference Equations20172017:186

https://doi.org/10.1186/s13662-017-1236-9

Received: 10 May 2017

Accepted: 12 June 2017

Published: 29 June 2017

Abstract

In this paper, we introduce a general \((p, q)\)-Sturm-Liouville difference equation whose solutions are \((p, q)\)-analogues of classical orthogonal polynomials leading to Jacobi, Laguerre, and Hermite polynomials as \((p, q) \to(1,1)\). In this direction, some basic characterization theorems for the introduced \((p, q)\)-Sturm-Liouville difference equation, such as Rodrigues representation for the solution of this equation, a general three-term recurrence relation, and a structure relation for the \((p, q)\)-classical polynomial solutions are given.

Keywords

\((p,q)\)-Sturm-Liouville problems \((p,q)\)-classical orthogonal polynomials \((p,q)\)-Pearson difference equation \((p,q)\)-integrals \((p,q)\)-difference operators

MSC

34B24 39A70

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 [13].

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, 812] 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
$$c_{0}(\alpha,\beta)= \int_{0}^{p/q} \frac{x^{\beta}}{((1,x p^{-2});(p,q))_{-\alpha}} \,d_{p,q}x. $$

Declarations

Acknowledgements

The authors thank both reviewers for their valuable comments. This work has been partially supported by the Agencia Estatal de Innovación (AEI) of Spain under grant MTM2016-75140-P, cofinanced by the European Community fund FEDER, and Xunta de Galicia, grants GRC 2015-004 and R 2016/022. F Soleyman thanks the hospitality of Departamento de Estatística, Análise Matemática e Optimización of Universidade de Santiago de Compostela, and Departamento de Matemática Aplicada II of Universidade de Vigo during her visits.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, K.N. Toosi University of Technology
(2)
E.E. Aeronáutica e do Espazo, Departamento de Matemática Aplicada II, Universidade de Vigo
(3)
Facultade de Matemáticas, Universidade de Santiago de Compostela

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