On Non-linear Characterizations of Classical Orthogonal Polynomials

Abstract

Classical orthogonal polynomials are known to satisfy seven equivalent properties, namely the Pearson equation for the linear functional, the second-order differential/difference/q-differential/ divided-difference equation, the orthogonality of the derivatives, the Rodrigues formula, two types of structure relations, and the Riccati equation for the formal Stieltjes function. In this work, following previous work by Kil et al. (J Differ Equ Appl 4:145–162, 1998a; Kyungpook Math J 38:259–281, 1998b), we state and prove a non-linear characterization result for classical orthogonal polynomials on non-uniform lattices. Next, we give explicit relations for some families of these classes.

1 Introduction

Univariate orthogonal polynomials (or orthogonal polynomials for short) are systems of polynomials $(p_n)_n$ with $\deg (p_n) = n$, satisfying a certain orthogonality relation. They are very useful in practice in various domains of mathematics, physics, engineering, image processing and so on, because of the many properties and relations they satisfy. As examples of areas where orthogonal polynomials play important roles, we could cite approximation theory (see for example [6, 31]) and also numerical analysis (see [14, 15]).

It is known that any family of orthogonal polynomials ${(p_n)_{n\ge 0}}$ satisfies a three-term recurrence relation of the form

$$\begin{aligned} {p_{n+1}(x)=(A_nx+B_n)p_n(x)-C_np_{n-1}(x),\;\;p_{-1}(x)=0.} \end{aligned}$$

(1.1)

If ${h_n=\langle {\mathcal {L}},p_n^2\rangle }$, where ${\mathcal {L}}$ is the corresponding linear functional with respect to the sequence ${(p_n)_{n\ge 0}}$ and $k_n$ is the leading coefficient of ${ p_n(x)}$ (see [17]), then

$$\begin{aligned} A_n=\frac{k_{n+1}}{k_n},\;\;C_n=\frac{A_n}{A_{n+1}} \frac{h_n}{h_{n+1}},\;\;n\ge 1 \end{aligned}$$

and we set $C_0=1$.

The systems of orthogonal polynomials associated with the names of Hermite, Laguerre, Jacobi and Bessel (including the special cases named after Tchebychev, Legendre, and Gegenbauer) are the most extensively and widely applied systems.

An orthogonal polynomial system $(p_n)_{n\ge 0}$ with respect to a weight function $\rho (x)$ is called classical if it satisfies one of the equivalent assertions (see [17]):

• $(p_n)_{n\ge 0}$ satisfies a second-order linear differential equation of the Sturm–Liouville type

  $$\begin{aligned} \phi (x)y''(x)+\psi (x)y'(x)+\lambda _n y(x)=0, \end{aligned}$$

  (1.2)

  where $\phi (x)$ is a polynomial of degree $\le 2$ and $\psi (x)$ is a polynomial of exact degree 1, both independent on n and $\lambda _n$ is independent on x.

• The derivatives $(p'_{n+1})_{n\ge 0}$ form an orthogonal polynomial system.

• The $p_n$s have the Rodrigues representation

  $$\begin{aligned} p_n(x)={\dfrac{D_n}{\rho (x)}}\left( \phi ^n(x)\rho (x)\right) ^{(n)},\quad n\ge 0. \end{aligned}$$

  (1.3)

• The weight function $\rho (x)$ satisfies a Pearson-type equation

  $$\begin{aligned} (\phi (x)\rho (x))'=\psi (x)\rho (x). \end{aligned}$$

  (1.4)

• The $p_n$s satisfy a difference-differential equation (or structure relation) of the form

  $$\begin{aligned} \pi (x)p'_n(x)=(\alpha _nx+\beta _n)p_n(x)+\gamma _n p_{n-1}(x). \end{aligned}$$

  (1.5)

In his paper [2], Al-Salam has obtained an expression for the derivative of the product of two consecutive Bessel polynomials and has shown that this expression does, in fact, characterize the Bessel polynomials. Based on this paper, McCarthy in [25] proved that there is an analogous characterization for very classical orthogonal polynomials (Hermite, Laguerre and Jacobi polynomials). This characterization can be stated as

• $(p_n)_{n\ge 0}$ satisfies a non-linear equation of the form:

  $$\begin{aligned} \phi (x)\dfrac{{\text {d}}}{{\text {d}}x}(p_n(x)p_{n-1}(x))=(\alpha _nx+\beta _n)p_n(x) p_{n-1}(x)+\gamma _n p_n^2(x)+\delta _n p^2_{n-1}(x){,} \end{aligned}$$

  (1.6)

  where $\alpha _n$, $\beta _n$, $\gamma _n$ and $\delta _n$ are independent on x.

Note that several other characterizations of classical orthogonal polynomials with respect to the derivative operator can be found in [23].

Very close to the very classical orthogonal polynomials (classical orthogonal polynomials of a continuous variable) are the classical orthogonal polynomials of a discrete variable. An orthogonal polynomial system $(p_n)_{n\ge 0}$ of a discrete variable with respect to a weight function $\rho (x)$ is called classical if it satisfies one of the equivalent assertions (see [1, 8, 13]):

• ${(p_n)_{n\ge 0}}$ satisfies a second-order linear difference equation of the Sturm–Liouville type

  $$\begin{aligned} \phi (x)\Delta \nabla y(x)+\psi (x)\Delta y(x)+\lambda _n y(x)=0, \end{aligned}$$

  (1.7)

  where $\phi (x)$ is a polynomial of degree $\le 2$ and $\psi (x)$ is a polynomial of exact degree 1, both independent on n and $\lambda _n$ is independent on x.

• The sequence of difference polynomials $(\Delta p_{n+1})_{n\ge 0}$ form an orthogonal polynomial system of discrete variable.

• The $p_n$s have the Rodrigues representation

  $$\begin{aligned} p_n(x)={\dfrac{D_n}{\rho (x)}}\Delta ^n\left( \phi ^n(x)\rho (x)\right) ,\quad n\ge 0. \end{aligned}$$

  (1.8)

• The weight function $\rho (x)$ satisfies a Pearson-type equation

  $$\begin{aligned} \Delta [\phi (x)\rho (x)]=\psi (x)\rho (x){.} \end{aligned}$$

  (1.9)

• The $p_n$s satisfy a difference equation (or structure relation) of the form

  $$\begin{aligned} \pi (x)\nabla p_n(x)=(\alpha _nx+\beta _n)p_n(x)+\gamma _n p_{n-1}(x){,} \end{aligned}$$

  (1.10)

  or otherwise stated (see [19])

  $$\begin{aligned} \phi (x) \nabla p_{n}(x)=\tilde{\alpha }_n p_{n+1}(x) +\tilde{\beta }_n p_n(x)+\tilde{\gamma }_n p_{n-1}(x){.} \end{aligned}$$

  (1.11)

• For each $n\ge 1$, $p_n$ and $p_{n-1}$ satisfy a relation of the form (see [21, Theorem 5.2])

  $$\begin{aligned}{} & {} \pi (x)\left[ p_n(x)\nabla p_{n-1}(x)+p_{n-1}(x)\nabla p_{n}(x)\right] \\{} & {} \quad =U_n p_n^2(x)+V_n p_{n-1}^2(x) +(W_nx+Y_n)p_n(x)p_{n-1}(x), \end{aligned}$$

  where the coefficients $U_n$, $V_n$, $W_n$ and $Y_n$ are independent on x and $\pi$ is a polynomial of degree less or equal to 2.

It should be noted that the operators $\Delta$ and $\nabla$ are respectively defined by

$$\begin{aligned} \Delta f(x)&= f(x+1)-f(x),\\ \nabla f(x)&= f(x)-f(x-1). \end{aligned}$$

Close to the classical discrete orthogonal polynomials are classical orthogonal polynomials of a q-discrete variable. An orthogonal polynomial system $(p_n)_{n\ge 0}$ of a q-discrete variable with respect to a weight function $\rho (x)$ is called classical if it satisfies one of the equivalent assertions (see [8, 18, 19]):

• ${(p_n)_{n\ge 0}}$ satisfies a second-order linear q-difference equation of the Sturm–Liouville type

  $$\begin{aligned} \phi (x){\mathcal {D}}_q{\mathcal {D}}_{\frac{1}{q}} y(x)+\psi (x){\mathcal {D}}_q y(x)+\lambda _n y(x)=0, \end{aligned}$$

  (1.12)

  where $\phi (x)$ is a polynomial of degree less than or equal to 2 and $\psi (x)$ is a polynomial of exact degree 1, both independent on n and $\lambda _n$ is independent on x.

• The sequence of q-difference polynomials $({\mathcal {D}}_q p_{n+1})_{n\ge 0}$ form an orthogonal polynomial system of a q-discrete variable.

• The $p_n$s have the Rodrigues representation

  $$\begin{aligned} p_n(x)={\dfrac{D_n}{\rho (x)}}{\mathcal {D}}_q^n\left( \phi ^n(x)\rho (x)\right) ,\quad n\ge 0. \end{aligned}$$

  (1.13)

• The weight function $\rho (x)$ satisfies a Pearson-type equation

  $$\begin{aligned} {\mathcal {D}}_q[\phi (x)\rho (x)]=\psi (x)\rho (x). \end{aligned}$$

  (1.14)

• The $p_n$s satisfy a q-difference equation (or structure relation) of the form (see [19])

  $$\begin{aligned} \phi (x) {\mathcal {D}}_{\frac{1}{q}} p_{n}(x)=\tilde{\alpha }_n p_{n+1}(x) +\tilde{\beta }_n p_n(x)+\tilde{\gamma }_n p_{n-1}(x). \end{aligned}$$

  (1.15)

• For each $n\ge 1$, $p_n$ and $p_{n-1}$ satisfy a relation of the form (see [22, Theorem 3.5])

  $$\begin{aligned}{} & {} \tilde{\pi }(x)\left[ p_n(x){\mathcal {D}}_{\frac{1}{q}} p_{n-1}(x)+p_{n-1}(x) {\mathcal {D}}_{\frac{1}{q}} p_{n}(x)\right] \\{} & {} \quad =\tilde{U}_n p_n^2(x)+\tilde{V}_n p_{n-1}^2(x) +(\tilde{W}_nx +\tilde{Y}_n)p_n(x)p_{n-1}(x), \end{aligned}$$

  where the coefficients $\tilde{U}_n$, $\tilde{V}_n$, $\tilde{W}_n$ and $\tilde{Y}_n$ are independent on x and $\tilde{\pi }$ is a polynomial of degree less than or equal to 2.

It should be noted that the q-derivative $D_q$ is defined as

$$\begin{aligned} D_q f(x)={\left\{ \begin{array}{ll} \dfrac{f(x)-f(qx)}{(1-q)x} &{} \text {if } q\ne 1 \text { and } x\ne 0\\ f'(0) &{}\text {if } x =0 \end{array}\right. }. \end{aligned}$$

The difference operator $\Delta$ and the q-derivative $D_q$ are both special cases of the Hahn’s operator $D_{q,\omega }$ (see [7]) which is defined as

$$\begin{aligned} D_{q,\omega } f(x)= \dfrac{f(qx+\omega )-f(x)}{(qx+\omega )-x}. \end{aligned}$$

More precisely, $D_q=D_{q,0}$ and $\Delta = D_{1,1}$.

In this paper, we prove equivalent non-linear characterization results similar to (1.6) for classical orthogonal polynomials on non-uniform lattices (including Wilson and Askey–Wilson polynomials). Also, we prove such a non-linear characterization for Meixner–Pollaczek and Continuous Hahn polynomials. Indeed, we give explicitly the coefficients of these relations for some families of classical orthogonal polynomials on non-uniform lattices.

2 Preliminaries

This section contains some preliminary definitions and results that are useful for a better reading of this article. The q-hypergeometric series, a fractional q-derivative and fractional q-integral are defined. The reader will consult the reference [18] for more informations about these concepts.

2.1 The Hypergeometric Series

In what follows, the symbol $(a)_n$ denotes the so-called Pochhammer symbol and is defined by

$$\begin{aligned} (a)_m=\left\{ \begin{array}{ll} 1 \quad \text {if }\;\; m=0 \\ a(a+1)\cdots (a+m-1)\quad \text {if }\;\; m=1,2,\ldots \\ \end{array}\right. \end{aligned}$$

and the hypergeometric series is defined as

$$\begin{aligned} {}_{p} F_{q}\left( \left. \begin{array}{c} a_1,\ldots , a_p \\ b_1,\ldots ,b_q \\ \end{array}\right| x\right) =\sum _{n=0}^{\infty }\frac{(a_1)_n\cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n}\frac{x^n}{n!}. \end{aligned}$$

2.2 The q-Hypergeometric Series

The basic hypergeometric or q-hypergeometric series $_r\phi _s$ is defined by the series

$$\begin{aligned} {}_{r} \phi _{s}\left( \left. \begin{array}{c} a_1,\ldots ,a_r\\ b_1,\ldots ,b_s \end{array}\right| q;z \right) :=\sum _{n=0}^\infty \frac{(a_1,\ldots ,a_r;q)_n}{(b_1,\ldots ,b_s;q)_n} \left( (-1)^n q^{\left( {\begin{array}{c}k\\ 2\end{array}}\right) }\right) ^{1+s-r}\frac{z^n}{(q;q)_n}, \end{aligned}$$

where

$$\begin{aligned} (a_1,\ldots ,a_r;q)_n:=(a_1;q)_n\cdots (a_r;q)_n, \end{aligned}$$

with

$$\begin{aligned} (a_i;q)_n=\left\{ \begin{array}{ll} \prod \limits _{j=0}^{n-1}(1-a_iq^j)&{} \text { if }\ n=1,2,3,\ldots \\ 1 &{} \text { if }\ n=0 \end{array}\right. . \end{aligned}$$

For $n=\infty$, we set

$$\begin{aligned} (a;q)_{\infty }=\prod _{n=0}^{\infty }(1-aq^n),\,\,|q|<1. \end{aligned}$$

The notation $(a;q)_n$ is the so-called q-Pochhammer symbol.

2.3 Difference and Divided-Difference Operators

2.3.1 The Operators ${\mathcal {D}}$ and ${\mathcal {S}}$

We define the difference operator ${\mathcal {D}}$ (see [26, 28]) and its companion operator ${\mathcal {S}}$ as follows:

$$\begin{aligned} {\mathcal {D}}f(x)=f\left( x+\frac{i}{2}\right) -f\left( x-\frac{i}{2}\right) , \quad {\mathcal {S}}f(x)=\frac{f\left( x+\frac{i}{2}\right) +f\left( x-\frac{i}{2}\right) }{2}, \end{aligned}$$

with $i^{2}=-1$.

The operator ${\mathcal {D}}$ transforms a polynomial of degree n ($n\ge 1$) in x into a polynomial of degree $n-1$ in x and a polynomial of degree 0 into the zero polynomial. The operator ${\mathcal {S}}$ transforms a polynomial of degree n in x into a polynomial of degree n in x.

The operators ${\mathcal {D}}$ and ${\mathcal {S}}$ fulfill the following properties.

Proposition 2.1

(See [26, 30]) The operators ${\mathcal {D}}$ and ${\mathcal {S}}$ satisfy the following product rules

$$\begin{aligned} {\mathcal {D}}(fg)= & {} {\mathcal {D}}f{\mathcal {S}}g+{\mathcal {S}}f {\mathcal {D}}g , \end{aligned}$$

(2.1)

$$\begin{aligned} {\mathcal {S}}(fg)= & {} \frac{1}{4}{\mathcal {D}}f{\mathcal {D}}g+ {\mathcal {S}}f {\mathcal {S}}g, \end{aligned}$$

(2.2)

$$\begin{aligned} {\mathcal {D}}{\mathcal {S}}= & {} {\mathcal {S}}{\mathcal {D}}, \end{aligned}$$

(2.3)

$$\begin{aligned} {\mathcal {S}}^2= & {} \frac{1}{4}{\mathcal {D}}^2+{\textbf{I}}, \end{aligned}$$

(2.4)

where ${\textbf{I}}f=f$.

2.3.2 The Operators ${\textbf{D}}$ and ${\textbf{S}}$

We define the difference operator ${\textbf{D}}$ (see [27]) and its companion operator ${\textbf{S}}$ as follows:

$$\begin{aligned} {\textbf{D}}f(x^{2})=\frac{f\left( (x+\frac{i}{2})^{2}\right) -f\left( (x-\frac{i}{2})^{2}\right) }{2ix},\;\; {\textbf{S}}f(x^{2}) =\frac{f\left( (x+\frac{i}{2})^{2}\right) +f\left( (x-\frac{i}{2})^{2}\right) }{2}, \end{aligned}$$

with $i^{2}=-1$. The operator ${\textbf{D}}$ transforms a polynomial of degree n ($n\ge 1$) in $x^2$ into a polynomial of degree $n-1$ in $x^2$ and a polynomial of degree 0 into the zero polynomial. The operator ${\textbf{S}}$ transforms a polynomial of degree n in $x^2$ into a polynomial of degree n in $x^2$.

The operators ${\textbf{D}}$ and ${\textbf{S}}$ fulfill the following properties.

Proposition 2.2

(See [27]) The operators ${\textbf{D}}$ and ${\textbf{S}}$ satisfy the following product rules

$$\begin{aligned} {\textbf{D}}(fg)= & {} {\textbf{D}}f{\textbf{S}}g+{\textbf{S}}f {\textbf{D}}g, \end{aligned}$$

(2.5)

$$\begin{aligned} {\textbf{S}}(fg)= & {} -x^2{\textbf{D}}f{\textbf{D}}g+ {\textbf{S}}f {\textbf{S}}g, \end{aligned}$$

(2.6)

$$\begin{aligned} {\textbf{D}}{\textbf{S}}= & {} {\textbf{S}}{\textbf{D}}-\frac{1}{2}{\textbf{D}}^2, \end{aligned}$$

(2.7)

$$\begin{aligned} {\textbf{S}}^2= & {} -x^2{\textbf{D}}^2-\frac{1}{2}{\textbf{S}}{\textbf{D}}+{\textbf{I}}, \end{aligned}$$

(2.8)

where $\textbf{I}f=f$.

2.3.3 The Operators ${\mathbb {D}}_{x}$ and ${\mathbb {S}}_{x}$

We define the operator ${\mathbb {D}}_{x}$ (called divided-difference operator) and its companion operator ${\mathbb {S}}_{x}$ (called mean operator) as [5, 9, 11, 29]

$$\begin{aligned} {\mathbb {D}}_{x}\,f(x(s))&={f(x(s+{1\over 2}))-f(x(s-{1\over 2})) \over x(s+{1\over 2})-x(s-{1\over 2}) },\\ {\mathbb {S}}_{x}\, f(x(s))&={ f(x(s+{1\over 2}))+f(x(s-{1\over 2})) \over 2}, \end{aligned}$$

where x(s) is a non-uniform lattice (see [9]). The operator ${\mathbb {D}}_{x}$ transforms a polynomial of degree n ($n\ge 1$) in x(s) into a polynomial of degree $n-1$ in x(s) and a polynomial of degree 0 into the zero polynomial. The operator ${\mathbb {S}}_{x}$ transforms a polynomial of degree n in x(s) into a polynomial of degree n in x(s).

The operators ${\mathbb {D}}_{x}$ and ${\mathbb {S}}_{x}$ satisfy the product rules

$$\begin{aligned} {\mathbb {D}}_{x}\left( f(x(s))g(x(s))\right)= & {} {\mathbb {S}}_{x}f(x(s)) \,{\mathbb {D}}_{x}g(x(s)) +{\mathbb {D}}_{x}f(x(s))\,{\mathbb {S}}_{x}g(x(s)), \end{aligned}$$

(2.9)

$$\begin{aligned} {\mathbb {S}}_{x}\left( f(x(s))g(x(s))\right)= & {} U_2(x(s))\,{\mathbb {D}}_{x}f(x(s))\,{\mathbb {D}}_{x}g(x(s))+{\mathbb {S}}_{x}f(x(s))\,{\mathbb {S}}_{x}g(x(s)),\nonumber \\ \end{aligned}$$

(2.10)

$$\begin{aligned} {\mathbb {D}}_{x}{\mathbb {S}}_{x}f= & {} \alpha \,{\mathbb {S}}_{x}{\mathbb {D}}_{x}f+U_{1}\,{\mathbb {D}}_{x}^2 f, \end{aligned}$$

(2.11)

$$\begin{aligned} {\mathbb {S}}_{x}^2f= & {} U_{1}{\mathbb {S}}_{x}\,{\mathbb {D}}_{x}f+\alpha U_{2}\,{\mathbb {D}}_{x}^2f+f, \end{aligned}$$

(2.12)

where $U_2$ is a polynomial of degree 2

$$\begin{aligned} U_2(x(s))=(\alpha ^2-1)\,x^2(s)+2\,\beta \,(\alpha +1)\,x(s)+\delta _{x}, \end{aligned}$$

(2.13)

and $\delta _{x}$ is a constant depending on $\alpha ,\,\beta$ and the initial values x(0) and x(1) of x(s):

$$\begin{aligned} \delta _{x}={x^2(0)+x^2(1)\over 4\alpha ^2}-{(2\alpha ^2-1)\over 2\alpha ^2}x(0)\,x(1) -{\beta \,(\alpha +1)\over \alpha ^2}(x(0)+x(1)) +{\beta ^2\,(\alpha +1)^2\over \alpha ^2}, \end{aligned}$$

and

$$\begin{aligned} U_1(s):= U_1(x(s))=(\alpha ^2-1)\,x(s)+\beta \,(\alpha +1), \;\;U_2(s):= U_2(x(s)). \end{aligned}$$

(2.14)

Note that

$$\begin{aligned} {\mathbb {D}}_{x}F_n(x(s))&=\gamma _n F_{n-1}(x(s)), \\ {\mathbb {S}}_{x}F_n(x(s))&=\alpha _n F_{n}(x(s))+\frac{\gamma _n}{2}\nabla x_{n+1}(\varepsilon ) F_{n-1}(x(s)), \end{aligned}$$

where $F_n(x(s))$ is a function defined in [24]. More properties of the non-uniform lattices x(s), the properties of the divided-difference operator ${\mathbb {D}}_{x}$ and its companion ${\mathbb {S}}_{x}$ can be found in [10,11,12, 16, 24] : x(s) satisfies the conditions

$$\begin{aligned} x(s+k)-x(s)= & {} \gamma _k \nabla x_{k+1}(s), \end{aligned}$$

(2.15)

$$\begin{aligned} {x(s+k)-x(s)\over 2}= & {} \alpha _{k} x_{k}(s)+\beta _{k}, \end{aligned}$$

(2.16)

for $k=0,1, \ldots$, with

$$\begin{aligned} \alpha _{0}=1, \alpha _{1}=\alpha , \beta _{0}=0, \beta _{1}=\beta , \gamma _{0}=0, \gamma _{1}=1, \end{aligned}$$

and the sequences $(\alpha _k)$, $(\beta _k)$, $(\gamma _k)$ satisfy the following relations

$$\begin{aligned}{} & {} \alpha _{k+1}-2\alpha \alpha _{k}+\alpha _{k-1}=0, \\{} & {} \beta _{k+1}-2\beta _{k}+\beta _{k-1}=2\beta \alpha _{k}, \\{} & {} \gamma _{k+1}-\gamma _{k-1}=2\alpha _{k}, \end{aligned}$$

for $k=0,1, \ldots$.

3 Non-linear Characterization for Meixner–Pollaczek and Continuous Hahn Polynomials

The Meixner–Pollaczek polynomials $P_n^{(\lambda )}(x;\varphi )$ and the Continuous Hahn polynomials $p_n(x;a,b,c,d)$, respectively, have the hypergeometric representation (see [18]):

$$\begin{aligned}{} & {} P_n^{(\lambda )}(x;\varphi )=\dfrac{(2\lambda )_n}{n!}e^{in\varphi } {}_{2} F_{1}\left( \left. \begin{array}{c} -n,\lambda +ix \\ 2\lambda \\ \end{array}\right| 1-e^{-2i\varphi }\right) , \end{aligned}$$

(3.1)

$$\begin{aligned}{} & {} \dfrac{p_n(x;a,b,c,d)}{(a+d)_n(a+c)_n}=\dfrac{i^n}{n!} {}_{3} F_{2}\left( \left. \begin{array}{c} -n,n+a+b+c+d-1,a+ix\\ a+c,a+d \\ \end{array}\right| 1\right) .\nonumber \\ \end{aligned}$$

(3.2)

They are known to satisfy the second-order difference equation (see [30])

$$\begin{aligned} \phi (x){\mathcal {D}}^2 y(x)+\psi (x) {\mathcal {S}}{\mathcal {D}}y(x)+\lambda _n y(x)=0, \end{aligned}$$

(3.3)

where $\phi$ and $\psi$ are polynomials of degree 2 and 1, respectively, and $\lambda$ is a constant depending on the degree of the polynomial solution and the parameters involved in the polynomials.

Note that for the Meixner–Pollaczek polynomials, we have (see [30])

$$\begin{aligned} \phi (x)= & {} i(\lambda \sin \varphi -x\cos \varphi ), \end{aligned}$$

(3.4)

$$\begin{aligned} \psi (x)= & {} 2(\lambda \cos \varphi +x\sin \varphi ), \end{aligned}$$

(3.5)

and

$$\begin{aligned} \lambda _n=-2in\sin \varphi , \end{aligned}$$

and for the Continuous Hahn polynomials we have (see [30])

$$\begin{aligned} \phi (x)= & {} -{x}^{2}+ \frac{i}{2}\left( a+b-c-d\right) x +\frac{1}{2}(ab+cd), \\ \psi (x)= & {} -i\left( a+b+c+d\right) x+cd-ab, \end{aligned}$$

and

$$\begin{aligned} \lambda =\lambda _n=-n(n+a+b+c+d-1). \end{aligned}$$

Theorem 3.1

(Non-linear characterization) Let ${(P_n)_{n\ge 0}}$ be a sequence of classical orthogonal polynomials on non-uniform lattice. Then, for $n\ge 1$, $P_n(x)$ and $P_{n-1}(x)$ satisfy

$$\begin{aligned}{} & {} \phi (x)\Big [P_{n}(x){\mathcal {S}}{\mathcal {D}}P_{n-1}(x)+P_{n-1}(x){\mathcal {S}}{\mathcal {D}}P_{n}(x)\Big ]\nonumber \\{} & {} \qquad +\psi (x)\Big [P_{n}(x){\mathcal {S}}^2P_{n-1}(x)+P_{n-1}(x){\mathcal {S}}^2P_{n}(x)\Big ]\nonumber \\{} & {} \quad =\left[ (\psi _1+2i\phi _2)x+\psi _0+\psi _1\left( {B_{n}\over 2A_{n}} -{B_{n-1}\over 2A_{n-1}}\right) \right. \nonumber \\{} & {} \qquad \left. +i\phi _2\left( n{B_{n}\over A_{n}} -(n-2){B_{n-1}\over A_{n-1}}\right) \right] P_{n}(x)P_{n-1}(x)\nonumber \\{} & {} \qquad +{1\over A_{n-1}}(\psi _1+(2n-3)i\phi _2)P_{n}^2(x) -{C_{n} \over A_{n}}(\psi _1+(2n-1)i\phi _2)P_{n-1}^2(x).\qquad \end{aligned}$$

(3.6)

Furthermore, if $(Q_n)_{n\in {\mathbb {N}}}$ is a sequence of polynomials such that $Q_0(x)=P_0(x)$ and, for $n\ge 1$, $Q_n(x)$ and $Q_{n-1}(x)$ satisfy (3.6). Then $Q_n(x)=P_n(x)$, for all $n\ge 0$.

Proof

Using the fact that the sequence $(P_n)_{{n\ge 0}}$ is a classical orthogonal polynomial sequence, for all non-negative integer n, $P_{n+1}(x)$ satisfies (3.3), namely:

$$\begin{aligned} \phi (x){\mathcal {D}}^2 P_{n+1}(x)+\psi (x) {\mathcal {S}}{\mathcal {D}}P_{n+1}(x)+\lambda _{n+1} P_{n+1}(x)=0, \end{aligned}$$

(3.7)

with

$$\begin{aligned} \phi (x)=\phi _2x^2+\phi _1x+\phi _0; \;\;\psi (x)=\psi _1x+\psi _0;\;\;\lambda _n=n(n-1)\phi _2-in\psi _1. \end{aligned}$$

In (1.1), using the relations (2.1), (2.2), (2.3) and (2.4), we obtain:

$$\begin{aligned} {\mathcal {D}}^2 P_{n+1}(x)=2iA_n{\mathcal {S}}{\mathcal {D}}P_n(x)+(A_nx+B_n){\mathcal {D}}^2P_n(x)-C_n{\mathcal {D}}^2P_{n-1}(x) \end{aligned}$$

(3.8)

and

$$\begin{aligned} {\mathcal {S}}{\mathcal {D}}P_{n+1}(x)=2iA_n{\mathcal {S}}^2 P_n(x)-iA_nP_n(x) +(A_nx+B_n){\mathcal {S}}{\mathcal {D}}P_n(x)-C_n{\mathcal {S}}{\mathcal {D}}P_{n-1}(x). \end{aligned}$$

(3.9)

Using (1.1), (3.8) and (3.9) to replace ${\mathcal {D}}^2 P_{n+1}(x)$, ${\mathcal {S}}{\mathcal {D}}P_{n+1}(x)$ and $P_{n+1}(x)$ in (3.7), we obtain:

$$\begin{aligned}{} & {} \phi (x){\mathcal {S}}{\mathcal {D}}P_{n}(x)+\psi (x){\mathcal {S}}^2P_{n}(x) =-{C_{n}\over 2iA_{n}}(\lambda _{n-1}-\lambda _{n+1})P_{n-1}(x)\nonumber \\{} & {} \qquad +\left[ \frac{1}{2}\psi (x) + \left( {1\over 2i}x+{B_{n}\over 2iA_{n}}\right) (\lambda _{n}-\lambda _{n+1}) \right] P_{n}(x). \end{aligned}$$

(3.10)

For $n\ge 2$, we replace n by $n-1$ in (3.10) and obtain:

$$\begin{aligned}{} & {} \phi (x){\mathcal {S}}{\mathcal {D}}P_{n-1}(x)+\psi (x){\mathcal {S}}^2P_{n-1}(x)\nonumber \\{} & {} \quad =-{C_{n-1}\over 2iA_{n-1}}(\lambda _{n-2}-\lambda _{n})P_{n-2}(x)\nonumber \\{} & {} \qquad +\left[ \frac{1}{2}\psi (x) + \left( {1\over 2i}x+{B_{n-1}\over 2iA_{n-1}}\right) (\lambda _{n-1}-\lambda _{n}) \right] P_{n-1}(x). \end{aligned}$$

(3.11)

We replace again n by $n-1$ in (1.1) and use the resulting relation to replace $P_{n-2}(x)$ in (3.11) to obtain:

$$\begin{aligned}{} & {} \phi (x){\mathcal {S}}{\mathcal {D}}P_{n-1}(x)+\psi (x){\mathcal {S}}^2P_{n-1}(x)\nonumber \\{} & {} \quad =-{1\over 2i A_{n-1}}(\lambda _{n-2}-\lambda _{n})P_{n}(x)\nonumber \\{} & {} \qquad +\left[ \frac{1}{2}\psi (x) + \left( {1\over 2i}x+{B_{n-1}\over 2iA_{n-1}}\right) (\lambda _{n-1}-\lambda _{n-2}) \right] P_{n-1}(x). \end{aligned}$$

(3.12)

If we multiply (3.10) by $P_{n-1}(x)$, (3.12) by $P_{n}(x)$ and add the resulting expression, we get:

$$\begin{aligned}&\phi (x)\left[ P_{n}(x){\mathcal {S}}{\mathcal {D}}P_{n-1}(x)+P_{n-1}(x){\mathcal {S}}{\mathcal {D}}P_{n}(x)\right] \\&\qquad +\psi (x)\left[ P_{n}(x){\mathcal {S}}^2P_{n-1}(x)+P_{n-1}(x){\mathcal {S}}^2P_{n}(x)\right] \\&\quad =\left[ (\psi _1+2i\phi _2)x+\psi _0+\psi _1\left( {B_{n}\over 2A_{n}}-{B_{n-1}\over 2A_{n-1}}\right) \right. \\&\qquad \left. +i\phi _2\left( n{B_{n}\over A_{n}}-(n-2){B_{n-1}\over A_{n-1}}\right) \right] P_{n}(x)P_{n-1}(x)\\&\qquad +{1\over A_{n-1}}(\psi _1+(2n-3)i\phi _2)P_{n}^2(x)-{C_{n}\over A_{n}}(\psi _1+(2n-1)i\phi _2)P_{n-1}^2(x). \end{aligned}$$

This proves the first part of Theorem 3.1.

Now, we prove the second part.

Let $(Q_n)_{n\in {\mathbb {N}}}$ be a sequence of polynomials of a quadratic variable such that $Q_0(x)=P_0(x)$ and, for $n\ge 1$, $Q_n(x)$ and $Q_{n-1}(x)$ satisfy

$$\begin{aligned}{} & {} \phi (x)\left[ Q_{n}(x){\mathcal {S}}{\mathcal {D}}Q_{n-1}(x)+Q_{n-1}(x){\mathcal {S}}{\mathcal {D}}Q_{n}(x)\right] \nonumber \\{} & {} \qquad +\psi (x)\left[ Q_{n}(x){\mathcal {S}}^2Q_{n-1}(x) +Q_{n-1}(x){\mathcal {S}}^2Q_{n}(x)\right] \nonumber \\{} & {} \quad =\left[ (\psi _1+2i\phi _2)x+\psi _0+\psi _1\left( {B_{n}\over 2A_{n}}-{B_{n-1}\over 2A_{n-1}}\right) +i\phi _2\left( n{B_{n}\over A_{n}}-(2-n){B_{n-1}\over A_{n-1}}\right) \right] \nonumber \\{} & {} \qquad Q_{n}(x)Q_{n-1}(x)\nonumber \\{} & {} \qquad +{1\over A_{n-1}}(\psi _1+(2n-3)i\phi _2)Q_{n}^2(x)-{C_{n}\over A_{n}}(\psi _1+(2n-1)i\phi _2)Q_{n-1}^2(x). \end{aligned}$$

(3.13)

Let $b_n$ be the leading coefficient of $Q_n(x)$. We shall firstly show by induction that $k_n=b_n$ for all $n\ge 0$. We have $b_0=k_0$ and we assume that $n\ge 1$ and $b_{n-1}=k_{n-1}$. If we compare the coefficients of $x^{2n}$ in (3.13), we find that we must consider two cases according as the degree of $\phi$ is less than two or equal to two.

• If the degree of $\phi$ is less than two then we have

  $$\begin{aligned} 2\psi _1b_nk_{n-1}=\psi _1b_nk_{n-1}+{\psi _1\over A_{n-1}}(b_n)^2 \end{aligned}$$

  and $b_n\ne 0$ implies that for the quadratic or q-quadratic variable, we have $b_n=A_{n-1}k_{n-1}=k_n$.

• If the degree of $\phi$ is equal to two, then we have

  $$\begin{aligned}&\phi _2\left( (n-1)ib_nk_{n-1}+nib_nk_{n-1}\right) +2\psi _1b_nk_{n-1}\\&\quad =(\psi _1+2i\phi _2)b_nk_{n-1}+{1\over A_{n-1}}\left( \psi _1+(2n-3)i\phi _2\right) b_n^2, \end{aligned}$$

  and the regularity of the corresponding linear functional with respect to the sequence ${(Q_n)_{n\ge 0}}$ implies that $\psi _1+(2n-3)i\phi _2\ne 0$ and $b_n\ne 0$ we have $b_n=A_{n-1}k_{n-1}=k_n$.

We have by assumption $Q_0(x)=P_0(x)$. Assume further that $n\ge 1$ and $Q_{n-1}(x)=P_{n-1}(x)$ but $Q_{n}(x)\ne P_{n}(x)$.

Then $Q_n(x)=P_n(x)+g(x)$ where $g(x)=c(x^r+\cdots )$, $c\ne 0$. Since $Q_n(x)$ and $P_n(x)$ have the same degree and the same leading coefficient, we must have $r<n$. From (3.13), we get

$$\begin{aligned}&\phi (x)\left[ (P_{n}(x)+g(x)){\mathcal {S}}{\mathcal {D}}P_{n-1}(x)+P_{n-1}(x){\mathcal {S}}{\mathcal {D}}(P_{n}(x)+g(x))\right] \\&\qquad +\psi (x)\left[ (P_{n}(x)+g(x)) {\mathcal {S}}^2P_{n-1}(x) +P_{n-1}(x){\mathcal {S}}^2(P_{n}(x)+g(x))\right] \\&\quad =\left[ (\psi _1+2i\phi _2)x+\psi _0+\psi _1 \left( {B_{n}\over 2A_{n}}-{B_{n-1}\over 2A_{n-1}}\right) \right. \\&\qquad \left. +i\phi _2\left( n{B_{n}\over A_{n}}-(2-n) {B_{n-1}\over A_{n-1}}\right) \right] (P_{n}(x)+g(x))P_{n-1}(x)\\&\qquad +{1\over A_{n-1}}(\psi _1+(2n-3)i\phi _2)(P_{n}(x)+g(x))^2\\ {}&\qquad -{C_{n} \over A_{n}}(\psi _1+(2n-1)i\phi _2)P_{n-1}^2(x). \end{aligned}$$

Using the fact that $P_n(x)$ and $P_{n-1}(x)$ satisfy (3.6) we obtain

$$\begin{aligned}{} & {} \phi (x)\left[ g(x){\mathcal {S}}{\mathcal {D}}P_{n-1}(x)+P_{n-1}(x){\mathcal {S}}{\mathcal {D}}g(x)\right] \nonumber \\{} & {} \qquad +\psi (x)\left[ g(x){\mathcal {S}}^2P_{n-1}(x)+P_{n-1}(x){\mathcal {S}}^2g(x)\right] \nonumber \\{} & {} \quad =\left[ (\psi _1+2i\phi _2)x+\psi _0+\psi _1\left( {B_{n}\over 2A_{n}}-{B_{n-1}\over 2A_{n-1}}\right) \right. \nonumber \\{} & {} \qquad \left. +i\phi _2\left( n{B_{n}\over A_{n}}-(2-n){B_{n-1}\over A_{n-1}}\right) \right] g(x)P_{n-1}(x)\nonumber \\{} & {} \qquad +{1\over A_{n-1}}(\psi _1+(2n-3)i\phi _2)(2P_{n}(x)g(x)+g(x)^2). \end{aligned}$$

(3.14)

We compare the coefficients of $x^{n+r}$ in (3.14). Let us consider two cases:

• 1. If the degree of $\phi$ is less than two, then we get

  $$\begin{aligned} 2\psi _1cb_{n-1}=\psi _1cb_{n-1}+2{b_n\over A_{n-1}}c\psi _1 \end{aligned}$$

  which is equivalent to

  $$\begin{aligned} 2\psi _1cb_{n-1}=\psi _1cb_{n-1}+2c\psi _1b_{n-1}. \end{aligned}$$

  Then, the fact that $\psi _1 b_{n-1}\ne 0$ implies that this is impossible if $c\ne 0$.

• 2. If the degree of $\phi$ is equal to two, then we get

  $$\begin{aligned}&\phi _2\left( ci(n-1)b_{n-1}+ricb_{n-1}\right) +2c\psi _1b_{n-1}\\&\quad =(\psi _1+2i\phi _2)cb_{n-1}+{2\over A_{n-1}}\left( \psi _1+(2n-3)i\phi _2\right) cb_n, \end{aligned}$$

  which is equivalent to

  $$\begin{aligned}&\phi _2\left( ci(n-1)b_{n-1}+ricb_{n-1}\right) +2c\psi _1b_{n-1}\\&\quad =(\psi _1+2i\phi _2)cb_{n-1}+2\left( \psi _1+(2n-3)i\phi _2\right) cb_{n-1}. \end{aligned}$$

  The regularity of the corresponding linear functional with respect to the sequence $(P_n)$ implies that $\psi _1+(3n-3-r)i\phi _2\ne 0$ and the previous equation is impossible if $c\ne 0$.

The proof is therefore completed. $\square$

The following corollaries give explicit coefficients for the non-linear characterization of the Meixner–Pollaczek and the Continuous Dual Hahn polynomials.

Corollary 3.2

The Meixner–Pollaczek polynomials are characterized by the following non-linear difference equation

$$\begin{aligned}&i(\lambda \sin \varphi -x\cos \varphi )\left[ P_{n}(x){\mathcal {S}}{\mathcal {D}}P_{n-1}(x)+P_{n-1}(x){\mathcal {S}}{\mathcal {D}}P_{n}(x)\right] \\&\qquad +2(\lambda \cos \varphi +x\sin \varphi )\left[ P_{n}(x){\mathcal {S}}^2P_{n-1}(x) +P_{n-1}(x){\mathcal {S}}^2P_{n}(x)\right] \\&\quad =\left[ 2\sin \varphi x+3\lambda \cos \varphi \right] P_{n}(x)P_{n-1}(x) +nP_{n}^2(x)-(n+2\lambda -1)P_{n-1}^2(x). \end{aligned}$$

Corollary 3.3

The Continuous Hahn polynomials are characterized by the following non-linear difference equation

$$\begin{aligned}&\left( -x^2+{i\over 2}(a+b-c-d)x+{1\over 2}(ab+cd)\right) \left[ P_{n}(x){\mathcal {S}}{\mathcal {D}}P_{n-1}(x)+P_{n-1}(x){\mathcal {S}}{\mathcal {D}}P_{n}(x)\right] \\&\qquad +(-i(a+b+c+d+2)x+cd-ab)\left[ P_{n}(x){\mathcal {S}}^2P_{n-1}(x) +P_{n-1}(x){\mathcal {S}}^2P_{n}(x)\right] \\&\quad = -{n(b+c+n-1)(b+d+n-1)\over 2n+a+b+c+d-2}P_{n-1}^2(x)\\&\qquad +\left( -i(a+b+c+d+2)x+D_n \right) P_{n}(x)P_{n-1}(x)\\&\qquad +\frac{(n-2+a+b+c+d)(n-1+a+c)(n-1+a+d)}{2n+a+b+c+d-2}P_{n}^2(x), \end{aligned}$$

where $D_n$ depends on n, a, b, c and d.

4 Non-linear Characterization for Wilson and Continuous Dual Hahn Polynomials

The Wilson polynomials $W_n(x^2;a,b,c,d)$ and Continuous Dual Hahn polynomials $S_n(x^2;a,b,c)$, respectively, have the hypergeometric representation (see [18]):

$$\begin{aligned} \frac{W_n(x^2;a,b,c,d)}{(a+b,a+c,a+d)_n}= & {} {}_{4} F_{3}\left( \left. \begin{array}{c} -n,n+a+b+c+d-1,a+ix,a-ix \\ a+b,a+c,a+d \\ \end{array}\right| 1\right) ,\nonumber \\ \end{aligned}$$

(4.1)

$$\begin{aligned} \frac{S_n(x^2;a,b,c)}{(a+b,a+c)_n}= & {} {}_{3} F_{2}\left( \left. \begin{array}{c} -n,a-ix,a+ix\\ a+b,a+c \\ \end{array}\right| 1\right) . \end{aligned}$$

(4.2)

They are known to satisfy the second-order divided-difference equation (see [27])

$$\begin{aligned} \phi (x^2){\textbf{D}}^2 y(x^2)+\psi (x^2) {\textbf{S}}{\textbf{D}}y(x^2)+\lambda _n y(x^2)=0, \end{aligned}$$

(4.3)

and these two families satisfy the three-term recurrence relation

$$\begin{aligned} P_{n+1}(x^2)=(A_nx^2+B_n)P_n(x^2)-C_nP_{n-1}(x^2),\;\;P_{-1}(x^2)=0. \end{aligned}$$

(4.4)

Theorem 4.1

(Non-linear characterization) Let $(P_n)_{n\in {\mathbb {N}}}$ be a sequence of classical orthogonal polynomials on a non-uniform lattice. Then, for $n\ge 1$, $P_n(x^2)$ and $P_{n-1}(x^2)$ satisfy

$$\begin{aligned}{} & {} \phi (x^2)\Big [P_{n}(x^2){\textbf{D}}{\textbf{S}}P_{n-1}(x^2)+P_{n-1}(x^2){\textbf{D}}{\textbf{S}}P_{n}(x^2)\Big ] \nonumber \\{} & {} \qquad +\psi (x^2)\Big [P_{n}(x^2){\textbf{S}}^2P_{n-1}(x^2) +P_{n-1}(x^2){\textbf{S}}^2P_{n}(x^2)\Big ]\nonumber \\{} & {} \quad =\left[ (\psi _1+2\phi _2)x^2+\psi _0+\psi _1\left( {B_{n} \over A_{n}}-{B_{n-1}\over A_{n-1}}\right) +\phi _2\left( n{B_{n} \over A_{n}}+(2-n){B_{n-1}\over A_{n-1}}\right) \right] \nonumber \\{} & {} \qquad \times P_{n}(x^2)P_{n-1}(x^2) +{1\over A_{n-1}}(\psi _1+(2n-3)\phi _2) P_{n}^2(x^2)\nonumber \\{} & {} \qquad -{C_{n}\over A_{n}}(\psi _1+(2n-1)\phi _2)P_{n-1}^2(x^2). \end{aligned}$$

(4.5)

Furthermore, if $(Q_n(x^2))_{n\in {\mathbb {N}}}$ is a sequence of polynomials such that $Q_0(x^2)=P_0(x^2)$ and, for $n\ge 1$, $Q_n(x)$ and $Q_{n-1}(x)$ satisfy (4.5). Then $Q_n(x^2)=P_n(x^2)$, for all $n\ge 0$.

Proof

For all integers n, $P_{n+1}(x^2)$ satisfies (4.3), namely:

$$\begin{aligned} \phi (x^2){\textbf{D}}^2 P_{n+1}(x^2)+\psi (x^2) {\textbf{S}}{\textbf{D}}P_{n+1}(x^2)+\lambda _{n+1} P_{n+1}(x^2)=0, \end{aligned}$$

(4.6)

with

$$\begin{aligned} \phi (x^2)=\phi _2x^4+\phi _1x^2+\phi _0; \;\;\psi (x^2)=\psi _1x^2+\psi _0;\;\;\lambda _n=-n(n-1)\phi _2-n\psi _1. \end{aligned}$$

From (4.4), using the relations (2.5), (2.6), (2.7) and (2.8), we obtain:

$$\begin{aligned} {\textbf{D}}^2 P_{n+1}(x^2)=2A_n{\textbf{D}}{\textbf{S}}P_n(x^2) +(A_nx^2+B_n){\textbf{D}}^2P_n(x^2)-C_n{\textbf{D}}^2P_{n-1}(x^2) \end{aligned}$$

(4.7)

and

$$\begin{aligned} {\textbf{S}}{\textbf{D}}P_{n+1}(x^2)=2A_n{\textbf{S}}^2 P_n(x^2)-A_nP_n(x^2) +(A_nx^2+B_n){\textbf{S}}{\textbf{D}}P_n(x^2)-C_n{\textbf{S}}{\textbf{D}}P_{n-1}(x^2). \end{aligned}$$

(4.8)

We use (4.4), (4.7) and (4.8) to replace ${\textbf{D}}^2 P_{n+1}(x^2)$, ${\textbf{S}}{\textbf{D}}P_{n+1}(x^2)$ and $P_{n+1}(x^2)$ in (4.6); we obtain:

$$\begin{aligned}{} & {} \phi (x^2){\textbf{D}}{\textbf{S}}P_{n}(x^2)+\psi (x^2){\textbf{S}}^2P_{n}(x^2)\nonumber \\{} & {} \quad =-{C_{n}\over 2A_{n}}(\lambda _{n-1}-\lambda _{n+1})P_{n-1}(x^2)\nonumber \\{} & {} \qquad +\left[ \frac{1}{2}\psi (x^2) + \left( {1\over 2}x^2+{B_{n}\over 2A_{n}}\right) (\lambda _{n}-\lambda _{n+1}) \right] P_{n}(x^2). \end{aligned}$$

(4.9)

For $n\ge 2$, we replace n by $n-1$ in (4.9) and we obtain:

$$\begin{aligned}{} & {} \phi (x^2){\textbf{D}}{\textbf{S}}P_{n-1}(x^2)+\psi (x^2){\textbf{S}}^2P_{n-1}(x^2)\nonumber \\{} & {} \quad =-{C_{n-1}\over 2A_{n-1}}(\lambda _{n-2}-\lambda _{n})P_{n-2}(x^2)\nonumber \\{} & {} \qquad +\left[ \frac{1}{2}\psi (x^2) + \left( {1\over 2}x^2+{B_{n-1}\over 2A_{n-1}}\right) (\lambda _{n-1}-\lambda _{n}) \right] P_{n-1}(x^2). \end{aligned}$$

(4.10)

We replace again n by $n-1$ in (4.4) and we use the resulting relation to replace $P_{n-2}(x^2)$ in (4.10) to obtain:

$$\begin{aligned}{} & {} \phi (x^2){\textbf{D}}{\textbf{S}}P_{n-1}(x^2)+\psi (x^2){\textbf{S}}^2P_{n-1}(x^2)\nonumber \\{} & {} \quad ={1\over 2 A_{n-1}}(\lambda _{n-2}-\lambda _{n})P_{n}(x^2)\nonumber \\{} & {} \qquad +\left[ \frac{1}{2}\psi (x^2) + \left( {1\over 2}x^2+{B_{n-1}\over 2A_{n-1}}\right) (\lambda _{n-1}-\lambda _{n-2}) \right] P_{n-1}(x^2). \end{aligned}$$

(4.11)

If we multiply (4.9) by $P_{n-1}(x^2)$, (4.11) by $P_{n}(x^2)$ and add the resulting expressions, we obtain:

$$\begin{aligned}&\phi (x^2)\left[ P_{n}(x^2){\textbf{D}}{\textbf{S}}P_{n-1}(x^2)+P_{n-1}(x^2){\textbf{D}}{\textbf{S}}P_{n}(x^2)\right] \\&\qquad +\psi (x^2)\left[ P_{n}(x^2){\textbf{S}}^2P_{n-1}(x^2)+P_{n-1}(x^2){\textbf{S}}^2P_{n}(x^2)\right] \\&\quad =\left[ (\psi _1+2\phi _2)x^2+\psi _0+\psi _1\left( {B_{n}\over A_{n}}-{B_{n-1}\over A_{n-1}}\right) +\phi _2\left( n{B_{n}\over A_{n}}+(2-n){B_{n-1}\over A_{n-1}}\right) \right] \\&\qquad \times P_{n}(x^2)P_{n-1}(x^2) +{1\over A_{n-1}}(\psi _1+(2n-3)\phi _2)P_{n}^2(x^2)\\ {}&\qquad -{C_{n}\over A_{n}}(\psi _1+(2n-1)\phi _2)P_{n-1}^2(x^2). \end{aligned}$$

This proves the first part of Theorem 4.1.

Now, we prove the second part.

Let $(Q_n)_{n\in {\mathbb {N}}}$ be a sequence of polynomials of a quadratic variable such that $Q_0(x^2)=P_0(x^2)$ and, for $n\ge 1$, $Q_n(x^2)$ and $Q_{n-1}(x^2)$ satisfy (4.5). Let $b_n$ be the leading coefficient of $Q_n(x^2)$. We shall first show by induction that $k_n=b_n$ for all $n\ge 0$. We have $b_0=k_0$ and we assume that $n\ge 1$ and $b_{n-1}=k_{n-1}$. If we compare the coefficients of $x^{4n}$ in (4.5), we find that we must consider two cases whether the degree of $\phi$ is less than two or equal to two.

• If the degree of $\phi$ is less than two then, we have

  $$\begin{aligned} 2\psi _1b_nk_{n-1}=\psi _1b_nk_{n-1}+{\psi _1\over A_{n-1}}(b_n)^2 \end{aligned}$$

  and $b_n\ne 0$ implies that for the quadratic or q-quadratic variable, we have $b_n=A_{n-1}k_{n-1}=k_n$.

• If the degree of $\phi$ is equal to two then, we have

  $$\begin{aligned}&\phi _2\left( (n-1)b_nk_{n-1}+nb_nk_{n-1}\right) +2\psi _1b_nk_{n-1} \\&\quad =(\psi _1+2\phi _2)b_nk_{n-1}+{1\over A_{n-1}}\left( \psi _1+(2n-3)\phi _2\right) b_n^2, \end{aligned}$$

  and the regularity of the corresponding linear functional with respect to the sequence $(Q_n)_{{n\ge 0}}$ implies that $\psi _1+(2n-3)\phi _2\ne 0$ and $b_n\ne 0$ we have $b_n=A_{n-1}k_{n-1}=k_n$.

We have by assumption $Q_0(x^2)=P_0(x^2)$. Assume further that $n\ge 1$ and $Q_{n-1}(x^2)=P_{n-1}(x^2)$ but $Q_{n}(x^2)\ne P_{n}(x^2)$.

Then $Q_n(x^2)=P_n(x^2)+g(x^2)$ where $g(x^2)=c(x^{2r}+\cdots )$, $c\ne 0$. Since $Q_n(x^2)$ and $P_n(x^2)$ have the same degree and the same leading coefficient, we must have $r<n$. From (4.5), we get

$$\begin{aligned}&\phi (x^2)\left[ (P_{n}(x^2)+g(x^2)){\textbf{D}}{\textbf{S}}P_{n-1}(x^2)+P_{n-1}(x^2){\textbf{D}}{\textbf{S}}(P_{n}(x^2)+g(x^2))\right] \\&\qquad +\psi (x^2)\left[ (P_{n}(x^2)+g(x^2)){\textbf{S}}^2P_{n-1}(x^2) +P_{n-1}(x^2){\textbf{S}}^2(P_{n}(x^2)+g(x^2))\right] \\&\quad =\left[ (\psi _1+2\phi _2)x^2+\psi _0+\psi _1\left( {B_{n}\over A_{n}}-{B_{n-1}\over A_{n-1}}\right) \right. \\&\qquad \left. +\phi _2\left( n{B_{n}\over A_{n}}+(2-n){B_{n-1}\over A_{n-1}}\right) \right] (P_{n}(x^2)+g(x^2))P_{n-1}(x^2)\\&\qquad +{1\over A_{n-1}}(\psi _1+(2n-3)\phi _2)(P_{n}(x^2)+g(x^2))^2\\ {}&\qquad -{C_{n}\over A_{n}}(\psi _1+(2n-1)\phi _2)P_{n-1}^2(x^2). \end{aligned}$$

Using the fact that $P_n(x)$ and $P_{n-1}(x)$ satisfy (4.5), we obtain

$$\begin{aligned}{} & {} \phi (x^2)\left[ g(x^2){\textbf{D}}{\textbf{S}}P_{n-1}(x^2)+P_{n-1}(x^2){\textbf{D}}{\textbf{S}}g(x^2)\right] \nonumber \\{} & {} \qquad +\psi (x^2)\left[ g(x^2){\textbf{S}}^2P_{n-1}(x^2)+P_{n-1}(x^2){\textbf{S}}^2g(x^2)\right] \nonumber \\{} & {} \quad =\left[ (\psi _1+2\phi _2)x^2+\psi _0+\psi _1\left( {B_{n}\over A_{n}}-{B_{n-1}\over A_{n-1}}\right) \right. \left. +\phi _2\left( n{B_{n}\over A_{n}}+(2-n){B_{n-1}\over A_{n-1}}\right) \right] \nonumber \\{} & {} \qquad \times g(x^2)P_{n-1}(x^2) +{1\over A_{n-1}}(\psi _1+(2n-3)\phi _2)(2P_{n}(x)g(x)+g(x)^2). \end{aligned}$$

(4.12)

We compare the coefficients of $x^{2n+2r}$ in (4.12) and consider two cases:

• 1. If the degree of $\phi$ is less than two, then

  $$\begin{aligned} 2\psi _1cb_{n-1}=\psi _1cb_{n-1}+2{b_n\over A_{n-1}}c\psi _1 \end{aligned}$$

  which is equivalent to

  $$\begin{aligned} 2\psi _1cb_{n-1}=\psi _1cb_{n-1}+2c\psi _1b_{n-1}. \end{aligned}$$

  Then, the fact that $\psi _1 b_{n-1}\ne 0$ implies that this is impossible if $c\ne 0$.

• 2. If the degree of $\phi$ is equal to two, then

  $$\begin{aligned}&\phi _2\left( c(n-1)b_{n-1}+rcb_{n-1}\right) +2c\psi _1b_{n-1}\\&\quad =(\psi _1+2\phi _2)cb_{n-1}+{2\over A_{n-1}}\left( \psi _1+(2n-3)\phi _2\right) cb_n \end{aligned}$$

  which is equivalent to

  $$\begin{aligned}&\phi _2\left( c(n-1)b_{n-1}+rcb_{n-1}\right) +2c\psi _1b_{n-1}\\&\quad =(\psi _1+2\phi _2)cb_{n-1}+2\left( \psi _1+(2n-3)\phi _2\right) cb_{n-1}. \end{aligned}$$

  The regularity of the corresponding linear functional with respect to the sequence $(P_n)_{{n\ge 0}}$ implies that $\psi _1+(3n-3-r)\phi _2\ne 0$ and the previous equality is impossible if $c\ne 0$.

$\square$

Corollary 4.2

The Wilson polynomials are characterized by the following non-linear difference equation

$$\begin{aligned}&\phi (x^2)\left[ P_{n}(x^2){\textbf{D}}{\textbf{S}}P_{n-1}(x^2)+P_{n-1}(x^2){\textbf{D}}{\textbf{S}}P_{n}(x^2)\right] \\&\qquad +\psi (x^2)\left[ P_{n}(x^2){\textbf{S}}^2P_{n-1}(x^2)+P_{n-1}(x^2){\textbf{S}}^2P_{n}(x^2)\right] \\&\quad =-\left[ {n(b+c+n-1)(b+d+n-1)(c+d+n-1)\over a+b+c+d+2n-2}\right] P_{n-1}^2(x^2)\\&\qquad +\left[ \frac{(a+b+c+d+n-2)(a+b+n-1)(a+c+n-1) (a+d+n-1)}{a+b+c+d+2n-2}\right] P_{n}^2(x^2)\\&\qquad +\left[ (a+b+c+d+2)x^2+D_n \right] P_{n}(x^2)P_{n-1}(x^2), \end{aligned}$$

where $D_n$ depends on n, a, b, c and d.

Corollary 4.3

The Continuous Dual Hahn polynomials are characterized by the following non-linear difference equation

$$\begin{aligned}&(-(a+b+c)x^2+abc)\left[ P_{n}(x^2){\textbf{D}}{\textbf{S}}P_{n-1}(x^2)+P_{n-1}(x^2){\textbf{D}}{\textbf{S}}P_{n}(x^2)\right] \\&\qquad +(x^2-ab-ac-bc)\left[ P_{n}(x^2){\textbf{S}}^2P_{n-1}(x^2)+P_{n-1} (x^2){\textbf{S}}^2P_{n}(x^2)\right] \\&\quad =\left( x^2+D_n\right) P_{n}(x^2)P_{n-1}(x^2) -(a+b+n-1)(a+c+n-1)P_{n}^2(x^2)\\&\qquad +n(b+c+n-1)P_{n-1}^2(x^2), \end{aligned}$$

where $D_n$ depends on n, a, b, c and d.

5 Non-linear Characterization for Orthogonal Polynomials on q-Quadratic Lattices

A family $p_n(x)$ of polynomials of degree n is a family of classical q-quadratic orthogonal polynomials (also known as orthogonal polynomials on non-uniform lattices) if it is the solution of a divided-difference equation of the type (see [8, 9])

$$\begin{aligned} \phi (x(s)){\mathbb {D}}_{x}^2y(x(s))+\psi (x(s)){\mathbb {S}}_{x}{\mathbb {D}}_{x}y(x(s))+\lambda _n y(x(s))=0, \end{aligned}$$

(5.1)

where $\phi$ is a polynomial of maximal degree two and $\psi$ is a polynomial of exact degree one, $\lambda _n$ is a constant depending on the integer n and the leading coefficients $\phi _2$ and $\psi _1$ of $\phi$ and $\psi$:

$$\begin{aligned} \lambda _n=-\gamma _n(\gamma _{n-1}\phi _2+\alpha _{n-1}\psi _1) \end{aligned}$$

and x(s) is a non-uniform lattice defined by

$$\begin{aligned} x(s)= c_1q^s+c_2q^{-s}+c_3, \quad c_1c_2\ne 0, \end{aligned}$$

(5.2)

and the sequences $(\alpha _n)$ and $(\gamma _n)$ are given explicitly by :

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

5.1 General Theorem

In this section, we state and prove a non-linear characterization result for classical orthogonal polynomials on non-uniform lattices. The result is stated in the following theorem.

Theorem 5.1

Let $(P_n)_{{n\ge 0}}$ be a sequence of classical orthogonal polynomials on a non-uniform lattice. Then, for $n\ge 1$, $P_n(x(s))$ and $P_{n-1}(x(s))$ satisfy

$$\begin{aligned}{} & {} \psi (x(s))\Big [P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s)) +P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\Big ]\nonumber \\{} & {} \qquad +\phi (x(s))\Big [P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\Big ]\nonumber \\{} & {} \quad = \Big [(D_{n-1}+D_{n}-G_{n-1}A_{n-1})x(s)+E_{n-1}\nonumber \\ {}{} & {} \qquad -G_{n-1}B_{n-1} +E_{n}\Big ]P_{n}(x(s))P_{n-1}(x(s))\nonumber \\{} & {} \qquad + G_{n-1}\left( P_{n}(x(s))\right) ^2 -C_{n}G_{n}\left( P_{n-1}(x(s))\right) ^2, \end{aligned}$$

(5.3)

where

$$\begin{aligned} D_{n}= & {} {1\over 2}(\lambda _n-\lambda _{n+1}+\psi _1),\quad E_{n}={1\over 2} \left( (\lambda _n-\lambda _{n+1}){B_{n}\over A_{n}}+\psi _0\right) ,\nonumber \\ G_{n}= & {} {1\over 2A_{n}}\left( \lambda _{n-1}-\lambda _{n+1}\right) . \end{aligned}$$

(5.4)

Furthermore, if $(Q_n)_{n\in {\mathbb {N}}}$ is a sequence of polynomials a on non-uniform lattice such that $Q_0(x)=P_0(x)$ and, for $n\ge 1$, $Q_n(x(s))$ and $Q_{n-1}(x(s))$ satisfy (5.3). Then $Q_n(x(s))=P_n(x(s))$, for all $n\ge 0$.

Proof

Using the fact that ${(P_n(x(s))_{n\ge 0}}$ is a classical q-orthogonal polynomial sequence on non-uniform lattice, substituting n by $n+1$ in (5.1) we obtain

$$\begin{aligned} \phi (x(s)){\mathbb {D}}_{x}^2P_{n+1}(x(s))+\psi (x(s)){\mathbb {S}}_{x}{\mathbb {D}}_{x}P_{n+1}(x(s)) +\lambda _{n+1}P_{n+1}(x(s))=0. \end{aligned}$$

(5.5)

In (1.1), using the product rules given in [12, page 407], in [11, pages 741-742] or in [10, page 4], we obtain:

$$\begin{aligned} {\mathbb {D}}_{x}^2 P_{n+1}(x(s))= & {} \left[ A_{n}\alpha ^2x(s)+A_{n}\beta (\alpha +1)+B_{n}\right] {\mathbb {D}}_{x}^2P_{n}(x(s))\nonumber \\{} & {} +2A_{n}{\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))-A_{n}U_1(x(s)){\mathbb {D}}_{x}^2P_{n}(x(s))\nonumber \\ {}{} & {} -C_{n}{\mathbb {D}}_{x}^2P_{n-1}(x(s)) \end{aligned}$$

(5.6)

and

$$\begin{aligned} {\mathbb {S}}_{x}{\mathbb {D}}_{x}P_{n+1}(x(s))= & {} \left[ A_{n}\alpha ^2x(s)+A_{n}\beta (\alpha +1)+B_{n}\right] {\mathbb {S}}_{x}{\mathbb {D}}_{x}P_{n}(x(s))\nonumber \\{} & {} +2A_{n}{\mathbb {S}}_{x}^2 P_{n}(x(s)) -A_{n}U_1(x(s)){\mathbb {S}}_{x}{\mathbb {D}}_{x}P_{n}(x(s))\nonumber \\{} & {} -A_{n} P_{n}(x(s))-C_{n}{\mathbb {S}}_{x}{\mathbb {D}}_{x}P_{n-1}(x(s)). \end{aligned}$$

(5.7)

Using (1.1), (5.6) and (5.7) to replace ${\mathbb {D}}_{x}^2 P_{n+1}(x(s))$ , ${\mathbb {S}}_{x}{\mathbb {D}}_{x}P_{n+1}(x(s))$ and $P_{n+1}(x(s))$ in (5.5), we obtain:

$$\begin{aligned}&\psi (x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))+\phi (x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\\&\quad =-{C_{n}\over 2A_{n}}\left[ \lambda _{n-1} -\lambda _{n+1}\right] P_{n-1}(x(s))\\&\qquad +{1\over 2}\left[ (\lambda _{n} -\lambda _{n+1}+\psi _1)x(s) +(\lambda _{n} -\lambda _{n+1}){B_{n}\over A_{n}} +\psi _0\right] P_{n}(x(s)),\,\forall n \ge 1, \end{aligned}$$

which is equivalent to

$$\begin{aligned}{} & {} \psi (x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))+\phi (x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\nonumber \\{} & {} \quad =\left[ D_{n}x(s)+E_{n}\right] P_{n}(x(s))-C_{n}G_{n}P_{n-1}(x(s)) ,\,\forall n \ge 1, \end{aligned}$$

(5.8)

where $D_{n}$, $E_n$ and $G_n$ are defined in (5.4). For $n\ge 2$, we replace n by $n-1$ in (5.8) and we obtain:

$$\begin{aligned}{} & {} \psi (x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s))+\phi (x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))\nonumber \\{} & {} \quad =\left[ D_{n-1}x(s)+E_{n-1}\right] P_{n-1}(x(s))-C_{n-1} G_{n-1}P_{n-2}(x(s)),\,\forall n \ge 2.\nonumber \\ \end{aligned}$$

(5.9)

We also replace n by $n-1$ in (1.1) and use the resulting relation to replace $P_{n-2}(x(s))$ in (5.9) to obtain:

$$\begin{aligned}{} & {} \psi (x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s))+\phi (x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))=G_{n-1}P_{n}(x(s))\nonumber \\{} & {} \quad +\left[ (D_{n-1}-G_{n-1}A_{n-1})x(s)+E_{n-1}-G_{n-1}B_{n-1}\right] P_{n-1}(x(s)) ,\,\forall n \ge 1.\nonumber \\ \end{aligned}$$

(5.10)

If we multiply (5.8) by $P_{n-1}(x(s))$ and (5.10) by $P_n(x(s))$ and add the resulting expressions, we obtain:

$$\begin{aligned}&\psi (x(s))\left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\phi (x(s))\left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad =\left[ (D_{n-1}-G_{n-1}A_{n-1}+D_{n})x(s)+E_{n-1}\right. \\ {}&\qquad \left. -G_{n-1}B_{n-1} +E_{n}\right] P_{n}(x(s))P_{n-1}(x(s))\\&\qquad + G_{n-1}\left( P_{n}(x(s))\right) ^2-C_{n}G_{n}\left( P_{n-1}(x(s))\right) ^2. \end{aligned}$$

This proves the first part of Theorem 5.1.

Now, we prove the second part.

Let $(Q_n(x(s)))_{n\in {\mathbb {N}}}$ be a sequence of polynomials of a q-quadratic variable such that $Q_0(x(s))=P_0(x(s))$ and, for $n\ge 1$, $Q_n(x(s))$ and $Q_{n-1}(x(s))$ satisfy

$$\begin{aligned}{} & {} \psi (x(s))\left[ Q_{n}(x(s)){\mathbb {S}}_{x}^2Q_{n-1}(x(s))+Q_{n-1}(x(s)) {\mathbb {S}}_{x}^2Q_{n}(x(s))\right] \nonumber \\{} & {} \qquad +\phi (x(s))\left[ Q_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}Q_{n-1}(x(s))+Q_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}Q_{n}(x(s))\right] \nonumber \\{} & {} \quad =\left[ (D_{n-1}-G_{n-1}A_{n-1}+D_{n})x(s) +E_{n-1}-G_{n-1}B_{n-1} +E_{n}\right] \nonumber \\{} & {} \qquad \times Q_{n}(x(s))Q_{n-1}(x(s)) +G_{n-1}\left( Q_{n}(x(s))\right) ^2-C_{n}G_{n}\left( Q_{n-1}(x(s))\right) ^2.\nonumber \\ \end{aligned}$$

(5.11)

Let $a_n$ be the leading coefficient of $Q_n(x(s))$. We shall firstly show by induction that $k_n=a_n$ for all $n\ge 0$. We have $a_0=k_0$ and we assume that $n\ge 1$ and $a_{n-1}=k_{n-1}$. If we compare the coefficients of $F_{2n}(x(s))$ in (5.11), we find that we must consider two cases whether the degree of $\phi$ is less than two or equal to two.

• If the degree of $\phi (x(s))$ is less than two then, we have

  $$\begin{aligned} \psi _1((\alpha _{n-1})^2+(\alpha _n)^2)a_nk_{n-1}=(D_n+D_{n-1}-A_{n-1}G_{n-1}) a_nk_{n-1}+G_{n-1}(a_n)^2 \end{aligned}$$

  and $a_n\ne 0$ implies that for the q-quadratic variable, we have $a_n=A_{n-1}k_{n-1}=k_n$.

• If the degree of $\phi (x(s))$ is equal to two then we get

  $$\begin{aligned}&\psi _1((\alpha _{n-1})^2+(\alpha _n)^2)a_nk_{n-1}+\phi _2(\alpha _{n-1}\gamma _{n-1} +\alpha _{n}\gamma _{n})a_nk_{n-1}\\&\quad =(D_n+D_{n-1}-A_{n-1}G_{n-1})a_nk_{n-1}+G_{n-1}(a_n)^2 \end{aligned}$$

  and $a_n\ne 0$ implies that for the quadratic case or the q-quadratic case, we have $a_n=A_{n-1}k_{n-1}=k_n$.

We have by assumption $Q_0(x(s))=P_0(x(s))$. Assume further that $n\ge 1$ and we $Q_{n-1}(x(s))=P_{n-1}(x(s))$ but $Q_{n}(x(s))\ne P_{n}(x(s))$.

Then $Q_n(x(s))=P_n(x(s))+g(x(s))$, where $g(x(s))=c(F_r(x(s))+\cdots )$, $c\ne 0$. Since $Q_n(x(s))$ and $P_n(x(s))$ have the same degree and the same leading coefficient, we must have $r<n$. From (5.11), we get

$$\begin{aligned}&\psi (x(s))\left[ (P_{n}(x(s))+g(x(s))){\mathbb {S}}_{x}^2P_{n-1}(x(s))\right. \\&\qquad \left. +P_{n-1}(x(s))({\mathbb {S}}_{x}^2P_{n}(x(s))+{\mathbb {S}}_{x}^2g(x(s)))\right] \\&\qquad +\phi (x(s))\left[ (P_{n}(x(s))+g(x(s))){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))\right. \\&\qquad \left. +P_{n-1}(x(s))({\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))+{\mathbb {D}}_{x}{\mathbb {S}}_{x}g(x(s)))\right] \\&\quad = \left[ (D_{n-1}-G_{n-1}A_{n-1}+D_{n})x(s)+E_{n-1}-G_{n-1}B_{n-1} +E_{n}\right] \\ {}&\qquad \times (P_{n}(x(s))P_{n-1}(x(s))+g(x(s))P_{n-1}(x(s))) +G_{n-1}\left( \left( P_{n}(x(s))\right) ^2\right. \\ {}&\qquad \left. +2g(x(s))P_{n}(x(s))+(g(x(s)))^2\right) -C_{n}G_{n}\left( P_{n-1}(x(s))\right) ^2. \end{aligned}$$

Using the fact that $P_n(x(s))$ and $P_{n-1}(x(s))$ satisfy (5.3), we obtain

$$\begin{aligned}{} & {} \psi (x(s))\left[ g(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s)) +P_{n-1}(x(s)){\mathbb {S}}_{x}^2g(x(s))\right] \nonumber \\{} & {} \qquad +\phi (x(s))\left[ g(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}g(x(s))\right] \nonumber \\{} & {} \quad =\left[ (D_{n-1}-G_{n-1}A_{n-1}+D_{n})x(s)+E_{n-1}\right. \nonumber \\{} & {} \qquad \left. -G_{n-1}B_{n-1} +E_{n}\right] g(x(s))P_{n-1}(x(s)))\nonumber \\{} & {} \qquad + G_{n-1}\left( 2g(x(s))P_{n}(x(s))+(g(x(s)))^2\right) . \end{aligned}$$

(5.12)

We compare the coefficients of $F_{n+r}(x(s))$ in (5.12). Two cases arise:

• 1. If the degree of $\phi (x(s))$ is less than two, then we get

  $$\begin{aligned} \psi _1\left( (\alpha _{n-1})^2+(\alpha _r)^2\right) c k_{n-1}=(D_n+D_{n-1}-G_{n-1}A_{n-1})c k_{n-1}+2c k_{n}G_{n-1}, \end{aligned}$$

  which is equivalent to

  $$\begin{aligned} \psi _1\left( (\alpha _{n-1})^2+(\alpha _r)^2\right) c k_{n-1}=(D_n+D_{n-1}-G_{n-1}A_{n-1})c k_{n-1}+2c k_{n-1}G_{n-1}A_{n-1}. \end{aligned}$$

  Then, for the q-quadratic variable, this is impossible if $c\ne 0$.

• 2. If the degree of $\phi (x(s))$ is equal to two, then we get

  $$\begin{aligned}&\psi _1\left( (\alpha _{n-1})^2+(\alpha _r)^2\right) c k_{n-1}+\phi _2\left( \alpha _{n-1}\gamma _{n-1}+\alpha _r\gamma _{r}\right) c k_{n-1}\\&\quad =(D_n+D_{n-1}-G_{n-1}A_{n-1})c k_{n-1}+2c k_{n}G_{n-1}, \end{aligned}$$

  which is equivalent to

  $$\begin{aligned}&\psi _1\left( (\alpha _{n-1})^2+(\alpha _r)^2\right) c k_{n-1}+\phi _2\left( \alpha _{n-1}\gamma _{n-1}+\alpha _r\gamma _{r}\right) c k_{n-1}\\&\quad =(D_n+D_{n-1}-G_{n-1}A_{n-1})c k_{n-1}+2c k_{n-1}G_{n-1}A_{n-1}. \end{aligned}$$

  Again this is impossible if $c\ne 0$.

$\square$

5.2 Special Cases

We can specialize the above result to the various classical orthogonal polynomials on non-uniform lattice, namely Askey–Wilson, q-Racah, Continuous dual q-Hahn, Continuous q-Hahn, Dual q-Hahn, Al-Salam Chihara, q-Meixner–Pollaczek, Continuous q-Jacobi, Dual q-Krawtchouk, Continuous big q-Hermite, Continuous q-Laguerre and Continuous q-Hermite polynomials. Note that the results for the Askey–Wilson and the q-Racah polynomials would be enough since the other families can be obtained by some limit transitions. But here, we would like to provide a complete database for all these polynomials orthogonal on a q-quadratic lattices.

5.2.1 Askey–Wilson Polynomials

The Askey–Wilson polynomials have the q-hypergeometric representation [18, P. 415]

$$\begin{aligned} \frac{a^np_n(x;a,b,c,d|q)}{(ab,ac,ad;q)_n} ={}_{4} \phi _{3}\left( \left. \begin{array}{c} q^{-n},abcdq^{n-1},ae^{i\theta },ae^{-i\theta } \\ ab,ac,ad \end{array}\right| q;q \right) , \quad x=\cos \theta . \end{aligned}$$

They satisfy the divided-difference equation (5.1) with

$$\begin{aligned} \phi \left( x(s) \right)= & {} 2\left( dcba+1 \right) x^2 \left( s \right) -\left( a+b+c+d+abc+abd+acd+bcd\right) x \left( s \right) \\{} & {} +\, ab+ac+ad+bc+bd+cd-abcd-1,\\ \psi \left( x(s) \right)= & {} {\frac{ 4\left( abcd-1 \right) q^{1\over 2}\,x \left( s \right) }{q-1}} +{\frac{2\left( a+b+c+d-abc-abd-acd-bcd\right) \, q^{1\over 2}}{q-1}}. \end{aligned}$$

The monic Askey–Wilson polynomials are characterized by the following non-linear recurrence relation

$$\begin{aligned}&\psi (x(s))\left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\phi (x(s))\left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s)) +P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad =\left[ K_{n}x(s)+M_n\right] P_{n}(x(s))P_{n-1}(x(s))\\&\qquad -{2(q+1)(abcdq^{n-{3\over 2}}-q^{{3\over 2}-n}) \over q-1}\left( P_{n}(x(s))\right) ^2\\&\qquad -{2(q+1)(abcdq^{n-{1\over 2}}-q^{{1\over 2}-n})\over q-1} C_{n}\left( P_{n-1}(x(s))\right) ^2, \end{aligned}$$

where

$$\begin{aligned} K_n= & {} {2\sqrt{q}[abcd\left( q^{2n-2}(q+1)^2+2q^n\right) -\left( 2q^n+(q+1)^2\right) ]\over (q-1)q^n}, \\ C_n= & {} {1\over 4} {(1- q^n)(1-abq^{n-1})(1-ac q^{n-1})(1-ad q^{n-1})(1-abcdq^{n-1})\over (1-abcd q^{2n-3})(1-abcd q^{2n-1})}\\{} & {} \times \frac{(1-bc q^{n-1})(1-bd q^{n-1})(1-cd q^{n-1})}{(1-abcd q^{2n-2})^2},\\ M_n= & {} \frac{2\sqrt{q}(q^{2n}-abcd)}{q^{n}(q-1)}\left( a+a^{-1}-(\widetilde{A_n} +\widetilde{C_n})\right) \\{} & {} +\left\{ \frac{2\sqrt{q}[q^{2n-1}-abcdq+(q+1)(q^{2n-2}-abcdq)]}{q^{n}(q-1)}\right\} \\{} & {} \times \left( a+a^{-1}-(\widetilde{A}_{n-1}+\widetilde{C}_{n-1})\right) \\{} & {} +\frac{2\left( a+b+c+d-abc-abd-acd-bcd\right) q^{1\over 2}}{q-1}, \end{aligned}$$

with

$$\begin{aligned} \widetilde{A_n}= & {} \frac{(1-abq^{n})(1-acq^{n})(1-adq^{n}) (1-abcdq^{n-1})}{a(1-abcdq^{2n-1})(1-abcdq^{2n})},\\ \widetilde{C_n}= & {} \frac{a(1-q^{n})(1-bcq^{n-1})(1-bdq^{n-1}) (1-cdq^{n-1})}{a(1-abcdq^{2n-1})(1-abcdq^{2n-2})}. \end{aligned}$$

5.2.2 q-Racah Polynomials

The q-Racah polynomials have the q-hypergeometric representation [18, P. 422]

$$\begin{aligned} R_n(\mu (x);\alpha ,\beta ,\gamma ,\delta |q) ={}_{4} \phi _{3}\left( \left. \begin{array}{c} q^{-n},\alpha \beta q^{n+1},q^{-x}, \delta \gamma q^{x+1}\\ \alpha q,\beta \delta q,\gamma q\end{array}\right| q;q \right) , n=0,1,2,\ldots ,N \end{aligned}$$

where

$$\begin{aligned} \mu (x):=q^{-x}+\delta \gamma q^{x+1} \end{aligned}$$

and

$$\begin{aligned} \alpha q=q^{-N}\quad \text {or}\quad \beta \delta q=q^{-N} \quad \text {or}\quad \gamma q=q^{-N}, \end{aligned}$$

with N a non-negative integer. They satisfy (5.1) with

$$\begin{aligned} \phi (x(s))= & {} \left( \beta \,\alpha \,{q}^{2}+1\right) x(s)^2 -q\left( \gamma \,q\alpha +\gamma \,q\beta \, \delta +q\alpha \,\beta \,\delta +q\alpha \,\beta +\beta \,\delta +\gamma \, \delta +\gamma +\alpha \right) x(s)\\{} & {} +2\,q \left( -{q}^{2}\alpha \,\beta \,\delta \,\gamma +\gamma \,q\alpha +{\gamma }^{2}q\delta +\gamma \,q{\delta }^{2}\beta +q\alpha \,\beta \,\delta +q\gamma \,\delta \,\alpha +\gamma \,q\beta \,\delta -\gamma \,\delta \right) ,\\ \psi (x(s))= & {} 2 \sqrt{q}\left( \,{\frac{ \left( \beta \,\alpha \,{q}^{2}-1 \right) }{q-1}}\right) x(s)\\{} & {} -2\,{\frac{{q}^{3/2} \left( \gamma \,q\alpha +\gamma \,q\beta \,\delta -\gamma \,\delta -\gamma +q\alpha \,\beta \,\delta +q\alpha \,\beta -\alpha -\beta \,\delta \right) }{q-1}}. \end{aligned}$$

The monic q-Racah polynomials are characterized by the following relation

$$\begin{aligned}&\psi (x(s))\left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s)) +P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\phi (x(s))\left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad = \left[ K_{n}x(s)+M_n\right] P_{n}(x(s))P_{n-1}(x(s)) -{(q+1)\sqrt{q}(\alpha \beta q^{2n}-q)\over (q-1)q^n}\left( P_{n}(x(s))\right) ^2 \\&\qquad -{(q+1)\sqrt{q}(\alpha \beta q^{2n+1}-1)\over (q-1)q^n}C_{n} \left( P_{n-1}(x(s))\right) ^2, \end{aligned}$$

where

$$\begin{aligned} K_n= & {} {\alpha \beta \sqrt{q}\left( q^n(q+1)^2+2q^2\right) \over q-1} -{\sqrt{q}\left( (q+1)^2+2q^n\right) \over q^n(q-1)}, \\ C_n= & {} {q(1-\alpha q^n)(1-\alpha \beta q^n)(1-\beta \delta q^n)(1-\gamma q^n) (1-q^n)(1-\beta q^n)(\gamma -\alpha \beta q^n)(\delta -\alpha q^n) \over (1-\alpha \beta q^{2n-1})(1-\alpha \beta q^{2n+1})(1-\alpha \beta q^{2n})^2},\\ M_n= & {} \frac{\sqrt{q}(\alpha \beta q^{2n}-1)}{q^{n}(q-1)}\left( \widetilde{A_n} +\widetilde{C_n}-q\gamma \delta -1 \right) \\{} & {} +\left( \frac{q^{\frac{3}{2}}(\alpha \beta q^{2n}-1) +\sqrt{q}(q+1)(\alpha \beta q^{2n}-q)}{q^{n}(q-1)}\right) \left( \widetilde{A}_{n-1}+\widetilde{C}_{n-1}-q\gamma \delta -1 \right) \\{} & {} -2\frac{\left( q\gamma \alpha +q\gamma \beta \delta -\gamma \delta -\gamma +q\alpha \beta \delta +q\alpha \beta -\alpha -\beta \delta \right) q^{3\over 2}}{q-1} \end{aligned}$$

with

$$\begin{aligned} \widetilde{A_n}= & {} \frac{(1-\alpha q^{n+1})(1-\alpha \beta q^{n+1}) (1-\beta \delta q^{n+1})(1-\gamma q^{n+1})}{(1-\alpha \beta q^{2n+1}) (1-\alpha \beta q^{2n+2})},\\ \widetilde{C_n}= & {} \frac{q(1-q^{n})(1-\beta q^{n})(\gamma -\alpha \beta q^{n}) (\delta -\alpha q^{n})}{(1-\alpha \beta q^{2n})(1-\alpha \beta q^{2n+1})}. \end{aligned}$$

5.2.3 Continuous Dual q-Hahn Polynomials

The Continuous Dual q-Hahn polynomials have the q-hypergeometric representation [18, P. 429]

$$\begin{aligned} \dfrac{a^np_n(x;a,b,c|q)}{(ab,ac;q)_n}= {}_{3} \phi _{2}\left( \left. \begin{array}{c} q^{-n}, ae^{i\theta },ae^{-i\theta }\\ ab,ac \end{array}\right| q,q \right) , \quad x=\cos \theta . \end{aligned}$$

They satisfy (5.1) with

$$\begin{aligned} \phi (x(s))= & {} 2(x(s))^2-(a+b+c+abc)x(s)-1+bc+ab+ac, \\ \psi (s(s))= & {} -{4\sqrt{q}\over q-1}x(s)+{2(a+b+c-abc)\sqrt{q}\over q-1}. \end{aligned}$$

The monic continuous dual q-Hahn polynomials are characterized by the following non-linear recurrence relation

$$\begin{aligned}&\left( -{4\sqrt{q}\over q-1}x(s)+{2(a+b+c-abc)\sqrt{q}\over q-1}\right) \left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\left( 2(x(s))^2-(a+b+c+abc)x(s)-1+bc+ab+ac\right) \\&\qquad \times \left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad = \left[ -\left( {2\left( (q+1)^2+2q^n\right) \sqrt{q}\over q^n(q-1)}\right) x(s)-{2q^{{1\over 2}}(B_n+(q^{2}+2q) B_{n-1})\over q^n(q-1)} +{2(a+b+c-abc)\sqrt{q}\over q-1} \right] \\&\qquad \times P_{n}(s)P_{n-1}(s) + {2(q+1)q^{{3\over 2}}\over q^n(q-1)} \left( P_{n}(x(s))\right) ^2+{2(q+1)q^{{1\over 2}}\over q^n(q-1)}C_n \left( P_{n-1}(x(s))\right) ^2, \end{aligned}$$

where

$$\begin{aligned} B_n= & {} -{1\over 2}\left( a+a^{-1}-a^{-1}(1-abq^n)(1-acq^n) -a(1-q^n)(1-bcq^{n-1})\right) \\ C_n= & {} {1\over 4}(1-q^n)(1-abq^{n-1})(1-acq^{n-1})(1-bcq^{n-1}). \end{aligned}$$

5.2.4 Continuous q-Hahn Polynomials

The Continuous q-Hahn polynomials have the q-hypergeometric representation [18, P. 415] or [8, P.  75]

$$\begin{aligned} \frac{(ae^{i\varphi })^{n}P_n(x;a,b,c,d;q)}{(ab,ac,ad;q)_n} ={}_{4} \phi _{3}\left( \left. \begin{array}{c} q^{-n},abcdq^{n-1},ae^{i(\theta +2\varphi )}, ae^{-i\theta } \\ abe^{2i\varphi },ac,ad \end{array}\right| q;q \right) , \end{aligned}$$

here $\displaystyle {x=\cos (\theta +\varphi )}$. They satisfy the divided-difference equation (5.1) with

$$\begin{aligned} \phi \left( x(s) \right)= & {} 2\left( dcba+1 \right) x^2 \left( s \right) -\frac{\left( d+dcb+at^{2}+bt^{2}ad+abct^{2}+c+acd+bt^{2}\right) x \left( s \right) }{t}\\{} & {} +\, \frac{cat^{2}+bt^{2}d-t^{2}cbad+cbt^{2}+cd+t^{2}+bt^{4}a+t^{2}ad}{t^{2}},\\ \psi \left( x(s) \right)= & {} {\frac{ 4\left( abcd-1 \right) q^{1\over 2}\,x \left( s \right) }{q-1}}-2\sqrt{q}\frac{(-c-d+cda-bt^{2}-at^{2}+dcb +cbat^{2}+bt^{2}q)}{(q-1)t}, \end{aligned}$$

where $t=e^{i\varphi }$.

The monic Continuous q-Hahn polynomials are characterized by the following non-linear recurrence relation

$$\begin{aligned}&\psi (x(s))\left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s))+P_{n-1}(x(s)) {\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\phi (x(s))\left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s)) +P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad =\left[ K_{n}x(s)+M_n\right] P_{n}(x(s))P_{n-1}(x(s))\\&\qquad + {2(q+1)(-abcdq^{2n-1}+q^{2})\over \sqrt{q}(q-1)q^n} \left( P_{n}(x(s))\right) ^2\\&\qquad + {2(q+1)(abcdq^{2n}-q)\over \sqrt{q}(q-1)q^n}C_{n} \left( P_{n-1}(x(s))\right) ^2, \end{aligned}$$

where

$$\begin{aligned} K_n= & {} {2 abcd[\left( q^{2n}(q+1)^2+2q^{n+2}\right) ]\over q^{\frac{3}{2}}(q-1)q^n} -{2\sqrt{q}\left( (q+1)^2+2q^{n}\right) \over (q-1)q^n}, \\ C_n= & {} {1\over 4} {(1- q^n)(1-bcq^{n-1})(1-bd q^{n-1})(1-cd q^{n-1}e^{-2i\varphi })(1-abcdq^{n-2})\over (1-abcd q^{2n-1})(1-abcd q^{2n-3})}\\{} & {} \times \frac{(1-ab q^{n-1}e^{2i\varphi })(1-ac q^{n-1}) (1-ad q^{n-1})}{(1-abcd q^{2n-2})^2},\\ M_n= & {} \frac{\sqrt{q}(1-abcd q^{2n})}{q^{n}(q-1)} \left( ae^{i\varphi }+a^{-1}e^{i\varphi }-(\widetilde{A_n}+\widetilde{C_n})\right) \\{} & {} -\left\{ (q+2)\frac{(q^{2}-q^{2n-1}abcd)}{\sqrt{q}(q-1)q^{n}}\right\} \left( ae^{i\varphi }+a^{-1}e^{i\varphi }-(\widetilde{A}_{n-1} +\widetilde{C}_{n-1})\right) \\{} & {} -2\sqrt{q}\frac{(-c-d+cda-bt^{2} -at^{2}+dcb+cbat^{2}+bt^{2}q)}{(q-1)t} \end{aligned}$$

with $t=e^{i\varphi }$,

$$\begin{aligned} \widetilde{A_n}= & {} \frac{(1-abq^{n}e^{2i\varphi })(1-acq^{n}) (1-adq^{n})(1-abcdq^{n-1})}{a e^{i\varphi }(1-abcdq^{2n-1})(1-abcdq^{2n})},\\ \widetilde{C_n}= & {} \frac{ae^{i\varphi }(1-q^{n})(1-bcq^{n-1}) (1-bdq^{n-1})(1-cdq^{n-1}e^{-2i\varphi })}{(1-abcdq^{2n-1})(1-abcdq^{2n-2})}. \end{aligned}$$

5.2.5 Dual q-Hahn Polynomials

The Dual q-Hahn polynomials have the q-hypergeometric representation ([18, P. 450] or [8, P. 76]

$$\begin{aligned} R_n(x(s);\gamma ,\delta , N|q)= {}_{3} \phi _{2}\left( \left. \begin{array}{c} q^{-n},q^{-s}, \gamma \delta q^{s+1}\\ \gamma q, q^{-N} \end{array}\right| q;q \right) , \quad n=0,1, \ldots , N, \end{aligned}$$

where $x(s)=q^{-s}+\gamma \delta q^{s+1}$ and N a non-negative integer. They satisfy (5.1) with

$$\begin{aligned} \phi (x(s))= & {} (x(s))^2-{(\gamma q +q^{N+1}\gamma \delta +\gamma q^{N+1}+1)x(s)\over 2q^{N}}\\{} & {} +{\gamma (q^{N+1}\gamma \delta -\delta q^{N}+\delta +1)q\over 2q^{N}}, \\ \psi (s(s))= & {} -{2\sqrt{q}\over q-1}x(s)+{(q^{N+1}\gamma \delta +\gamma q^{N+1}-\gamma q+1)\sqrt{q}\over (q-1)q^{N}}. \end{aligned}$$

The monic Dual q-Hahn polynomials are characterized by the following non-linear recurrence relation

$$\begin{aligned}&\psi (x(s))\left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s)) +P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\phi (x(s))\left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s)) +P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad = \left[ K_{n}x(s)+M_n\right] P_{n}(x(s)) P_{n-1}(x(s))+{(q+1)q^{\frac{3}{2}}\over (1-q)q^n}\left( P_{n}(x(s))\right) ^2 \\&\qquad + {(1+q)\sqrt{q}\over (1-q)q^{n}}C_n\left( P_{n-1}(x(s))\right) ^2, \end{aligned}$$

where

$$\begin{aligned} K_n= & {} {\sqrt{q}\left( 2q^n-q^2+1\right) \over q^{n}(1-q)}, \\ M_n= & {} {\sqrt{q}(-q^{2}\gamma \delta +2\gamma \delta q-q^{2}\gamma +2\gamma q-2q^{-N+1}\gamma +q^{-N}\gamma -q^{N+1}+2q^{-N} )\over q-1},\\ C_n= & {} \gamma q(1-q^{n-N-1})(1-\gamma q^{n})(1-q^{n})(\delta -q^{n-N-1}). \end{aligned}$$

5.2.6 Al-Salam–Chihara Polynomials

The Al-Salam–Chihara polynomials have the q-hypergeometric representation [18, P. 455] or [8, P. 77]

$$\begin{aligned} Q_n(x; a,b |q)= {(ab; q)_n\over a^{n}} {}_{3} \phi _{2}\left( \left. \begin{array}{c} q^{-n},ae^{i\theta }, ae^{-i\theta } \\ ab, 0 \end{array}\right| q;q \right) , x=\cos \theta . \end{aligned}$$

They satisfy the divided-difference equation (5.1) with

$$\begin{aligned} \phi (x(s))= & {} 2(x(s))^2-(a+b)x(s)+ab-1,\\ \psi (s(s))= & {} -{4\sqrt{q}x(s)\over q-1}+{2(a+b)\sqrt{q}\over q-1}. \end{aligned}$$

The monic Al-Salam–Chihara polynomials are characterized by the following non-linear recurrence relation

$$\begin{aligned}&\psi (x(s))\left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s)) +P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\phi (x(s))\left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad = \left[ K_{n}x(s)+M_n\right] P_{n}(x(s))P_{n-1}(x(s)) +{2(q+1)q^{\frac{3}{2}}\over (1-q)q^n}\left( P_{n}(x(s))\right) ^2 \\&\qquad + {(1+q)(1-q^{n})(1-ab q^{n-1})\over 2(1-q)q^{n-\frac{1}{2}}} \left( P_{n-1}(x(s))\right) ^2, \end{aligned}$$

where

$$\begin{aligned} K_n= & {} {2\sqrt{q}\left( 2q^n-q^2+1\right) \over q^{n}(1-q)}, \\ M_n= & {} {(a+b)\sqrt{q}(q-3)\over 1-q}. \end{aligned}$$

5.2.7 q-Meixner–Pollaczek Polynomials

The q-Meixner–Pollaczek polynomials [18, P. 460] or [8, P. 78]

$$\begin{aligned} P_n(x;a|q)=a^{-n}e^{-in\varphi }\frac{(a^{2}; q)_n}{(q;q)_n} {}_{3} \phi _{2}\left( \left. \begin{array}{c} q^{-n},ae^{i(\theta +2\varphi )},ae^{-i\theta }\\ a^{2},0 \end{array}\right| q,q \right) , x=\cos (\theta +\varphi ). \end{aligned}$$

They satisfy (5.1) with

$$\begin{aligned} \phi (x(s))= & {} 2(x(s))^2-2a \cos \varphi x(s)+a^{2}-1, \\ \psi (s(s))= & {} -{4\sqrt{q}\over q-1}x(s)+{4a\sqrt{q}\cos \varphi \over q-1}. \end{aligned}$$

The monic q-Meixner–Pollaczek polynomials are characterized by the following non-linear recurrence relation

$$\begin{aligned}&\left( -{4\sqrt{q}\over q-1}x(s)+{4a\sqrt{q}\cos \varphi \over q-1}\right) \left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\left( 2(x(s))^2-2a \cos \varphi x(s)+a^{2}-1\right) \\&\qquad \times \left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad = \left[ -\left( {2\left( (q+1)^2+2q^n\right) \sqrt{q}\over q^n(q-1)}\right) x(s) +\frac{2a\sqrt{q}\cos \varphi (q+5)}{q-1} \right] P_{n}(x(s))P_{n-1}(x(s))\\&\qquad + {2(q+1)q^{{3\over 2}}\over q^n(q-1)}\left( P_{n}(x(s))\right) ^2 +{\left( {\sqrt{q}(q+1)(1-q^{n})(1-a^{2}q^{n-1})\over 2q^n(q-1)}\right) } \left( P_{n-1}(x(s))\right) ^2. \end{aligned}$$

5.2.8 Continuous q-Jacobi Polynomials

The Continuous q-Jacobi polynomials have the q-hypergeometric representation [18, P. 463] or [8, P. 78]

$$\begin{aligned}{} & {} P^{(\alpha , \beta )}_n(x|q)\\ {}{} & {} \quad =\frac{(q^{\alpha +1};q)_n}{(q;q)_n} {}_{4} \phi _{3}\left( \left. \begin{array}{c} q^{-n},q^{n+\alpha +\beta +1},q^{\frac{1}{2}\alpha +\frac{1}{4}}e^{i\theta },q^{\frac{1}{2}\alpha +\frac{1}{4}}e^{-i\theta } \\ q^{\alpha +1},-q^{\frac{1}{2}(\alpha +\beta +1)}, -q^{\frac{1}{2}(\alpha +\beta +2)} \end{array}\right| q;q \right) , x=\cos \theta . \end{aligned}$$

They satisfy the divided-difference equation (5.1) with

$$\begin{aligned} \phi \left( x(s) \right)= & {} \left( p^{2\alpha +2\beta +4}+1 \right) x^2 \left( s \right) + \frac{1}{2}(p+1)p^{\frac{1}{2}} \left( p^{2\alpha +2\beta +2}-p^{\alpha }-p^{\alpha +2\beta +2} +p^{\beta }\right) x \left( s \right) \\{} & {} -\frac{1}{2}\left( p^{2\alpha +2\beta +4}+p^{\alpha +\beta +3} -p^{2\alpha +2}+p^{\alpha +\beta +2}-p^{2\beta +2}+p^{\alpha +\beta +1}+1\right) ,\\ \psi \left( x(s) \right)= & {} {\frac{4p\left( p^{2\alpha +2\beta +4}-1 \right) \,x \left( s \right) }{(p-1)(p+1)}}-{\frac{ \left( -p^{2\alpha +\beta +2} -p^{\alpha }+p^{\alpha +2\beta +2}+p^{\beta }\right) \, p^{3\over 2}}{p-1}}, \end{aligned}$$

with $p=q^{2}$.

The monic Continuous q-Jacobi polynomials are characterized by the following non-linear recurrence relation

$$\begin{aligned}&\psi (x(s))\left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s))+P_{n-1} (x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\phi (x(s))\left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad = \left[ K_{n}x(s)+M_n\right] P_{n}(x(s))P_{n-1}(x(s))\\&\qquad - {q^{{3\over 2}}(q+1)(q^{\alpha +\beta +2n-1}-1) \over (q-1)q^{n}}\left( P_{n}(x(s))\right) ^2\\&\qquad - {\sqrt{q}(q+1)(q^{\alpha +\beta +2n+1}-1)\over (q-1)q^{n}} C_{n}\left( P_{n-1}(x(s))\right) ^2, \end{aligned}$$

where

$$\begin{aligned} K_n= & {} {\sqrt{q}[\left( (q+1)^2\left( -1+q^{\alpha +\beta +2}\right) \right) +\left( 2q^n\left( -1+q^{\alpha +\beta +2}\right) \right) ]\over (q-1)q^n}, \\ C_n= & {} {1\over 4} {(1-q^{n})(1- q^{n+\alpha })(1- q^{n+\beta }) (1-q^{n+\alpha +\beta })\left( 1-q^{n+\frac{1}{2}(\alpha +\beta -1)}\right) \over (1-q^{2n-1+\alpha +\beta })(1- q^{2n+1+\alpha +\beta })}\\{} & {} \times \frac{\left( 1+q^{n+\frac{1}{2}(\alpha +\beta +1)}\right) \left( 1-q^{n+\frac{1}{2}(\alpha +\beta )}\right) ^{2}}{(1- q^{2n+\alpha +\beta })^2},\\ M_n= & {} -{\sqrt{q}(q+1)(q^{\alpha +\beta +2n+2}-1)\over 2(q-1)q^{n}} \left( q^{\frac{1}{2}\alpha +\frac{1}{4}}+q^{-\frac{1}{2}\alpha -\frac{1}{4}}-(\widetilde{A_n}+\widetilde{C_n})\right) \\{} & {} -\left\{ \frac{q^{\frac{3}{2}}[q^{\alpha +\beta +2n}-1+(q+1) (q^{\alpha +\beta +2n+-1}-1)]}{2q^{n}(q-1)}\right\} \\{} & {} \times \left( q^{\frac{1}{2}\alpha +\frac{1}{4}}+q^{-\frac{1}{2} \alpha -\frac{1}{4}}-(\widetilde{A}_{n-1}+\widetilde{C}_{n-1})\right) \\{} & {} +{\frac{\left( -p^{2\alpha +\beta +2}-p^{\alpha }+p^{\alpha +2\beta +2} +p^{\beta }\right) \, p^{3\over 2}}{p-1}}, \end{aligned}$$

with

$$\begin{aligned} \widetilde{A_n}= & {} \frac{(1-q^{n+\alpha +1})(1-q^{\alpha +\beta +n+1}) (1-q^{n+\frac{1}{2}(\alpha +\beta +1)})(1-q^{n+\frac{1}{2}(\alpha +\beta +2)})}{q^{\frac{1}{2}\alpha +\frac{1}{4}}(1-q^{\alpha +\beta +2n+1}) (1-q^{\alpha +\beta +2n+2})},\\ \widetilde{C_n}= & {} \frac{q^{\frac{1}{2}\alpha +\frac{1}{4}}(1-q^{n}) (1-q^{n+\beta })\left( 1+q^{n+\frac{1}{2}(\alpha +\beta )}\right) \left( 1+q^{n+\frac{1}{2}(\alpha +\beta +1)}\right) }{(1-q^{\alpha +\beta +2n})(1-q^{\alpha +\beta +2n+1})}. \end{aligned}$$

5.2.9 Dual q-Krawtchouk Polynomials

The Dual q-Krawtchouk polynomials [18, P. 505] or [8, P. 80]

$$\begin{aligned} K_n(x(s);c,N|q)= {}_{3} \phi _{2}\left( \left. \begin{array}{c} q^{-n},q^{-s},cq^{s-N} \\ q^{-N},0 \end{array}\right| q,q \right) , \quad n=0,1, \ldots , N, \end{aligned}$$

where $x(s)=q^{-s}+cq^{s-N}$. They satisfy (5.1) with

$$\begin{aligned} \phi (x(s))= & {} (x(s))^2-(c+1)q^{-N}x(s)-2c(q^{-N}-q^{-2N}), \\ \psi (s(s))= & {} -{2\sqrt{q}\over q-1}x(s)+{2(c+1)\sqrt{q}\over (q-1)q^{N}}. \end{aligned}$$

The monic Dual q-Krawtchouk polynomials are characterized by the following non-linear recurrence relation

$$\begin{aligned}&\left( -{2\sqrt{q}\over q-1}x(s)+{2(c+1)\sqrt{q}\over (q-1)q^{N}}\right) \left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\left( (x(s))^2-(c+1)q^{-N}x(s)-2c(q^{-N}-q^{-2N})\right) \\&\qquad \times \left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))+P_{n-1}(x(s)) {\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad = \left[ -\left( {\left( 2q^n+(1+q)^2\right) \sqrt{q}\over q^n(q-1)}\right) x(s) +\frac{(c+1)\sqrt{q}(q+5)}{(q-1)q^{N}} \right] P_{n}(x(s))P_{n-1}(x(s))\\&\qquad + {(q+1)q^{{3\over 2}}\over q^n(q-1)}\left( P_{n}(x(s))\right) ^2 +{\left( {c\sqrt{q}(q+1)(1-q^{n})(1-q^{n-N-1})\over q^{n+N}(q-1)}\right) } \left( P_{n-1}(x(s))\right) ^2. \end{aligned}$$

5.2.10 Continuous Big q-Hermite Polynomials

The continuous big q-Hermite polynomials [18, P. 509]

$$\begin{aligned} H_n(x;a,|q)=a^{-n} {}_{3} \phi _{2}\left( \left. \begin{array}{c} q^{-n},ae^{i\theta },ae^{-i\theta } \\ 0,0 \end{array}\right| q,q \right) , \quad x=\cos \theta . \end{aligned}$$

They satisfy (5.1) with

$$\begin{aligned} \phi (x(s))= & {} 2(x(s))^2-ax(s)-1, \\ \psi (s(s))= & {} -{4\sqrt{q}\over q-1}x(s)+{2a\sqrt{q}\over q-1}. \end{aligned}$$

The Continuous big q-Hermite polynomials are characterized by the following non-linear recurrence relation

$$\begin{aligned}&\left( -{4\sqrt{q}\over q-1}x(s)+{2a\sqrt{q}\over q-1}\right) \left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\left( 2(x(s))^2-ax(s)-1\right) \left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad = \left[ -\left( {2\left( (q+1)^2+2q^n\right) \sqrt{q} \over q^n(q-1)}\right) x(s)+\frac{a \sqrt{q}(q+5)}{q-1} \right] P_{n}(x(s))\\&\qquad \times P_{n-1}(x(s)) + {2(q+1)q^{{3\over 2}}\over q^n(q-1)} \left( P_{n}(x(s))\right) ^2-{(q+1)(q^n-1)\sqrt{q}\over 2q^n(q-1)} \left( P_{n-1}(x(s))\right) ^2. \end{aligned}$$

5.2.11 Continuous q-Laguerre Polynomials

The Continuous q-Laguerre polynomials have the q-hypergeometric representation [18, P. 514] or [8, P. 81]

$$\begin{aligned} P^{(\alpha )}_n(x|q)={(q^{\alpha +1};q)_n\over (q;q)_n} {}_{3} \phi _{2}\left( \left. \begin{array}{c} q^{-n},q^{\frac{1}{2}\alpha +\frac{1}{4}}e^{i\theta },q^{\frac{1}{2}\alpha +\frac{1}{4}}e^{-i\theta }\\ q^{\alpha +1},0 \end{array}\right| q;q \right) , x=\cos \theta . \end{aligned}$$

They satisfy the divided-difference equation (5.1) with

$$\begin{aligned} \phi (x(s))= & {} 2(x(s))^2-{p^{\alpha +{1\over 2}}(p+1)x(s) \over 2q^{N}}+p^{2\alpha +2}-1, \\ \psi (s(s))= & {} -{4p\over p^{2}-1}x(s)+{p^{\alpha +\frac{3}{2}}\over (p-1)}, \end{aligned}$$

with $p=q^{2}$.

The monic Continuous q-Laguerre polynomials are characterized by the following non-linear recurrence relation

$$\begin{aligned}&\psi (x(s))\left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s)) +P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\phi (x(s))\left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s)) +P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad =\left[ K_{n}x(s)+M_n\right] P_{n}(x(s))P_{n-1}(x(s))\\&\qquad - G_{n-1}\left( P_{n}(x(s))\right) ^2-G_{n}C_{n}\left( P_{n-1}(x(s))\right) ^2, \end{aligned}$$

where

$$\begin{aligned} \xi _n K_n&= q^{n^{2}+2}-q^{11}+q^{10}+q^{n^{2}+5}+q^{9} -pq^{\frac{15}{2}}+2pq^{\frac{13}{2}}-q^{n^{2}+4}-q^{8}-2pq^{\frac{9}{2}}\\&\quad -pq^{n^{2}+\frac{3}{2}} -q^{n^{2}+3}-q^{n^{2}+6}+pq^{\frac{7}{2}} +q^{7}-q^{5}-q^{6}+q^{4}-q^{n^{2}+9}\\&\quad +q^{n^{2}+8}+pq^{n^{2}+\frac{11}{2}}+4pq^{n+\frac{11}{2}} -2pq^{n^{2}+\frac{9}{2}}-8pq^{n+\frac{9}{2}}\\&\quad +q^{n^{2}+7}+4pq^{n+\frac{7}{2}}+2pq^{n^{2}+\frac{5}{2}}, \end{aligned}$$

with $\xi _n=-(q-1)^{2}(p^{2}-1)q^{n+\frac{7}{2}}$,

$$\begin{aligned} \nu _n M_n&= q^{2n+\frac{1}{2}\alpha +\frac{1}{2}} +q^{2n+\frac{1}{2}\alpha +1}+q^{2n+\frac{1}{2}\alpha +\frac{3}{2}}+q^{2n+\frac{1}{2}\alpha +2}+q^{\frac{1}{2}\alpha +\frac{7}{2}} +q^{\frac{1}{2}\alpha +4}\\&\quad +q^{2n+\frac{1}{2}\alpha +3}+q^{2n+\frac{1}{2}\alpha +\frac{5}{2}} 4q^{\alpha +\frac{3}{2}}p^{\frac{11}{4}}+4q^{\alpha +\frac{5}{2}}p^{\frac{11}{4}}-q^{2n+\frac{1}{2}\alpha +3}p^{2} -q^{2n+\frac{1}{2}\alpha +\frac{5}{2}}p^{2}\\&\quad -q^{2n+\frac{1}{2}\alpha +\frac{1}{2}}p^{2}-q^{2n+\frac{1}{2}\alpha +\frac{1}{2}}p +q^{2n+\frac{1}{2}\alpha +\frac{7}{2}}p +q^{2n+\frac{1}{2}\alpha +3}p-q^{2n+\frac{1}{2}\alpha +2}p^{2}\\&\quad -q^{2n+\frac{1}{2}\alpha +\frac{3}{2}}p^{2}-q^{2n+\frac{1}{2}\alpha +1} p^{2}-q^{2n+\frac{1}{2}\alpha }p-q^{\frac{1}{2}\alpha +\frac{9}{2}}p -q^{\frac{1}{2}\alpha +4}p^{2}\\&\quad -q^{\frac{1}{2}\alpha +4}p-q^{\frac{1}{2}\alpha +\frac{7}{2}}p^{2} +q^{\frac{1}{2}\alpha +\frac{7}{2}}p+q^{\frac{1}{2}\alpha +3}p, \end{aligned}$$

with $\nu _n=2(p^{2}-1)q^{\frac{11}{4}}$,

$$\begin{aligned} \tau _n G_n&= q^{n^{2}+\frac{7}{2}}+q^{\frac{9}{2}}p-pq^{n^{2} +\frac{5}{2}}-pq^{\frac{7}{2}}-q^{n^{2}+7}-pq^{n^{2}+\frac{3}{2}} -pq^{\frac{5}{2}}+q^{8}+pq^{n^{2}+5} \\&\quad +q^{n^{2}+3}+pq^{n^{2}+\frac{1}{2}}+pq^{\frac{3}{2}} -q^{6}-q^{4}+q^{2}-q^{n^{2}+1}, \end{aligned}$$

where $\tau _n=-(q-1)^{2}(p^{2}-1)q^{n+\frac{3}{2}}$,

$$\begin{aligned} C_n= & {} {1\over 4} (1-q^{n})(1- q^{n+\alpha }). \end{aligned}$$

5.2.12 Continuous q-Hermite Polynomials

The Continuous q-Hermite polynomials have the q-hypergeometric representation [18, P. 540] or [8, P. 82]

$$\begin{aligned} H_n(x|q)=e^{in\theta } {}_{2} \phi _{0}\left( \left. \begin{array}{c} q^{-n},0 \\ - \end{array}\right| q;q^{n} e^{-2i n \theta } \right) , x=\cos \theta . \end{aligned}$$

They satisfy the divided-difference equation (5.1) with

$$\begin{aligned} \phi (x(s))= & {} 2(x(s))^2-1, \\ \psi (s(s))= & {} -{4\sqrt{q}\over q-1}. \end{aligned}$$

The monic Continuous q-Hermite polynomials are characterized by the following non-linear recurrence relation

$$\begin{aligned}&\psi (x(s))\left[ P_{n}(x(s)){\mathbb {S}}_{x}^2P_{n-1}(x(s)) +P_{n-1}(x(s)){\mathbb {S}}_{x}^2P_{n}(x(s))\right] \\&\qquad +\phi (x(s))\left[ P_{n}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n-1}(x(s))+P_{n-1}(x(s)){\mathbb {D}}_{x}{\mathbb {S}}_{x}P_{n}(x(s))\right] \\&\quad = \left[ {\sqrt{q}\left( 2q^n-q^2+1\right) x(s) \over q^{n}(1-q)}\right] P_{n}(x(s))P_{n-1}(x(s)) +{(1+q)q^{\frac{3}{2}}\over (1-q)q^n}\left( P_{n}(x(s))\right) ^2 \\&\qquad - {(1+q)\sqrt{q}\over 2(1-q)q^{n}}\left( P_{n-1}(x(s))\right) ^2. \end{aligned}$$