On the q-translation associated with the Askey-Wilson operator

Abstract

In this article, we solve the open problem 24.5.6 given in the study of Ismail, which consists of extending the action of q-translation operators introduced by Ismail to some measurable functions by means of basic Fourier theory. Also, we prove that the q-exponential function is the only solution of the q-analogue of the Cauchy functional equation. As application we give an inversion formula for the q-Gauss Weierstrass transform.

2000 Mathematics Subject Classification: 33D45; 33D60.

Introduction

The concept of the q-translation operators $E_q^y$ introduced by Ismail [1] was defined in polynomials through their action on the continuous q-Hermite polynomials H _m (x | q) as follows

$$E_q^y{H}_n\left(x|q\right)=\sum _{m=0}^n\frac{{\left(q;q\right)}_n}{{\left(q;q\right)}_m{\left(q;q\right)}_{n-m}}{H}_m\left(x|q\right)g_{n-m}\left(y\right)q^{\left(m^2-n^2\right)/4},$$

(1)

where

$$g_n\left(cos\left(\theta \right)\right)=q^{n^2/4}\left(1+e^{2i\theta }\right)e^{-in\theta }{\left(-q^{2-n}{e}^{2i\theta };q^2\right)}_{n-1}.$$

In others words

$$\frac{q^{n^2/4}}{{\left(q;q\right)}_n}{E}_q^y{H}_n\left(x|q\right)=\sum _{0\le m,j,m+2j\le n}\frac{q^{j+\left(m^2+{\left(n-m-2j\right)}^2\right)/4}}{{\left(q^2;q^2\right)}_j}\frac{H_m\left(x|q\right)H_{n-m-2j}\left(y|q\right)}{{\left(q;q\right)}_m{\left(q;q\right)}_{n-m-2j}},$$

where the polynomials H _n (x | q) are defined by (see [2, 3])

$$H_n\left(\text{cos}\theta |q\right)=\sum _{k=0}^n\frac{{\left(q;q\right)}_n}{{\left(q;q\right)}_{n-k}{\left(q;q\right)}_k}{e}^{i\left(n-2k\right)\theta }.$$

(2)

It was shown in [1] that the q-translation operators $E_q^y$ commute with the Askey-Wilson operator ${\mathcal{D}}_q$ on the space of all polynomials and by the use of the following expansion [3]

$${\left(q{\alpha }^2;q^2\right)}_{\infty }{\mathcal{E}}_q\left(x;\alpha \right)=\sum _{n=0}^{\infty }\frac{{\alpha }^n}{{\left(q;q\right)}_n}{q}^{n^2/4}{H}_n\left(x|q\right).$$

(3)

In ([1], (2.21)), the author proved the following product formula for the q-exponential function

$$E_q^y{\mathcal{E}}_q\left(x;\alpha \right)={\mathcal{E}}_q\left(x;\alpha \right){\mathcal{E}}_q\left(y;\alpha \right).$$

(4)

Furthermore, if y and z are two complex variables, then we have [1]

$$\begin{array}{c}{E}_q^y{E}_q^z=E_q^z{E}_q^y,\\ E_q^{y}f\left(x\right)=E_q^{x}f\left(y\right).\end{array}$$

In [3] problem 24.5.6, Ismail proposed the extension of the action of $E_q^y$ to measurable functions and proving that the only measurable functional solution of the q-analogue of the Cauchy functional equation

$$E_q^{y}f\left(x\right)=f\left(x\right)f\left(y\right)$$

is the q-exponential function. ${\mathcal{E}}_q\left(x;\alpha \right)$. The purpose of this paper is to define a new q-translation operator $T_q^x$ related to Askey-Wilson operator acting in some measurable functions by means of the basic Fourier series. We show that the new q-Translation coincides with $E_q^y$ on the set of continuous q-Hermite polynomials. In the same context, we establish many properties satisfied by the q-translation operator and generalizing the classical ones.

In the first section, we recall some results of basic Fourier series given in [4]. In Section "Preliminaries", we define and study the q-translation operator $T_q^x$. Also we solve the following problem

$$\left\{\begin{array}{c}\hfill {\mathcal{D}}_{q,x}u\left(x,y\right)={\mathcal{D}}_{q,y}u\left(x,y\right)\hfill \\ \hfill u\left(x,0\right)=f\left(x\right),f\in H_{\epsilon }.\hfill \end{array}\right.$$

(5)

As a consequence of (5) we solve the basic analogue of the Cauchy functional equation

$$f\left(x\oplus y\right):=T_q^{x}f\left(y\right)=f\left(x\right)f\left(y\right),$$

(6)

where the function f is in the same subspace of L²(w (x) dx). In addition, we prove the q-translation invariance of the measure w (x) dx over (− 1, 1). Some q-analogous of the Gauss Weierstrass transforms are studied in Section "q-Gauss Weierstrass transform".

Preliminaries

Let 0 < q < 1 and a ∈ $ℂ$, the q-shift factorial is defined by (see [2])

$$\begin{array}{c}{\left(a;q\right)}_0=1,\\ {\left(a;q\right)}_n={\prod }_{k=0}^{n-1}\left(1-a{q}^k\right),n=1,2,\dots ,\infty .\end{array}$$

Given a function f (x) with x = cos θ, f (x) can be viewed as a function of e^iθ.

Let

$$\stackrel{⌣}{f}\left(e^{i\theta }\right):=f\left(\text{cos}\theta \right).$$

The Askey-Wilson-divided difference operator ${\mathcal{D}}_q$ is defined by

$$\left({\mathcal{D}}_{q}f\right)\left(x\right)=\frac{\stackrel{⌣}{f}\left(q^{\frac{1}{2}}{e}^{i\theta }\right)-\stackrel{⌣}{f}\left(q^{-\frac{1}{2}}{e}^{i\theta }\right)}{\left(q^{\frac{1}{2}}-q^{{-}^{\frac{1}{2}}}\right)i\text{sin}\theta }.$$

(7)

The q-exponential function is given by [5]

$$\begin{array}{ll}\hfill {\mathcal{E}}_q\left(\text{cos}\theta ,\text{cos}\varphi ;\alpha \right)& =\frac{{\left({\alpha }^2;q^2\right)}_{\infty }}{{\left(q{\alpha }^2;q^2\right)}_{\infty }}\sum _{n=0}^{\infty }{\left(-e^{i\left(\theta +\varphi \right)}{q}^{\left(1-n\right)/2},-e^{i\left(\theta -\varphi \right)}{q}^{\left(1-n\right)/2};q\right)}_n\\ \times \frac{{\left(\alpha e^{-i\varphi }\right)}^n}{{\left(q;q\right)}_n}{q}^{n^2/4}.\end{array}$$

The q-exponential function ${\mathcal{E}}_q\left(\text{cos}\theta ,\text{cos}\varphi ;\alpha \right)$ is a solution of the q-difference equation of first-order [3]

$${\mathcal{D}}_q{\mathcal{E}}_q\left(\text{cos}\theta ,\text{cos}\varphi ;\alpha \right)=\frac{2\alpha q^{1/4}}{1-q}{\mathcal{E}}_q\left(\text{cos}\theta ,\text{cos}\varphi ;\alpha \right).$$

Put

$${\mathcal{E}}_q\left(x;\alpha \right)={\mathcal{E}}_q\left(x,0;\alpha \right),$$

then, we have (see [3])

$${\mathcal{E}}_q\left(x,y;\alpha \right)={\mathcal{E}}_q\left(x;\alpha \right){\mathcal{E}}_q\left(y;\alpha \right),$$

and

$$\underset{q\to 1}{\text{lim}}{\mathcal{E}}_q\left(x;\left(1-q\right)\alpha \right)=\text{exp}\left(2\alpha x\right).$$

Ismail and Zhang [5, 3] defined the q-cosine and q-sine functions through their q-exponential function as in the standard way, i.e.,

$${\mathcal{E}}_q\left(x;i\omega \right)=C_q\left(x;\omega \right)+i{S}_q\left(x;\omega \right),$$

and used transformation formulas to continue them analytically to entire functions in the variable ω. Bustoz and Suslov [4] have established the following orthogonality relations

$${\int }_{-1}^1{\mathcal{E}}_q\left(x;i{\omega }_n\right)\overline{{\mathcal{E}}_q\left(x;i{\omega }_m\right)}w\left(x\right)dx=2k\left({\omega }_n\right){\delta }_{n,m},$$

(8)

where

$$\begin{array}{ll}\hfill w\left(\text{cos}\theta \right)& =\frac{{\left(e^{2i\theta },e^{-2i\theta };q\right)}_{\infty }}{{\left(q^{1/2}{e}^{2i\theta },q^{1/2}{e}^{-2i\theta };q\right)}_{\infty }},\\ \hfill k\left(\omega \right)& =\pi \frac{{\left(q^{1/2};q\right)}_{\infty }^2}{{\left(q;q\right)}_{\infty }^2}\frac{{\left(-{\omega }^2;q^2\right)}_{\infty }}{{\left(-q{\omega }^2;q^2\right)}_{\infty }}\sum _{k=0}^{\infty }\frac{q^{k/2}}{1+{\omega }^2{q}^k},\end{array}$$

ω₀ = 0, ω₁ < ω₂ < ..., are zeros of the q-sine function S _q ((q^{1/ 4}+ q^{-1/ 4})/2); ω) and for n = 1, 2, . . . , ω _-n = −ω _n .

From [4], we have the following asymptotic estimates as n → ∞

$${\omega }_n~q^{1/4-n},andk\left({\omega }_n\right)~2\pi \frac{{\left(-q;q^2\right)}_{\infty }}{{\left(-q^{1/2};q\right)}_{\infty }^2},$$

(9)

and for 0 < ε < 1/ 2 and |x| ≤ 1 < (q^ε + q^-ε)/ 2, we have

$$\left|{\mathcal{E}}_q\left(x;i{\omega }_n\right)\right|<\frac{{\left(-q^{1/4-\epsilon }\left|{\omega }_n\right|;q^{1/2}\right)}_{\infty }}{{\left(q,q^{2\epsilon },q^{1-2\epsilon };q\right)}_{\infty }{\left(-q{\omega }_n^2;q^2\right)}_{\infty }}~C{q}^{-2\epsilon n},$$

(10)

where C = 1/(−q^{1/ 2}, q; q)_∞(q^ε, q^{1/ 2-ε}; q^{1/ 2})_∞.

q-Translation

We define the q-Fourier transform ${\mathcal{F}}_q$ as

$${\mathcal{F}}_q\left(f\right)\left(n\right)={\int }_{-1}^1{\mathcal{E}}_q\left(x;-i{\omega }_n\right)f\left(x\right)w\left(x\right)dx,n\in \mathbb{Z}.$$

Put

$$l^2\left(k\left({\omega }_n\right)\right)=\left\{{\left(z_n\right)}_{n\in \mathbb{Z}};\sum _{n=-\infty }^{\infty }\frac{1}{2k\left({\omega }_n\right)}|z_n{|}^2<\infty \right\}.$$

Theorem 1. The transform${\mathcal{F}}_q$ is an isomorphism from L²((− 1, 1), w(x)dx) into l² (k (ω _n )) and its inverse is given by

$$f\left(x\right)=\sum _{n=-\infty }^{\infty }\frac{1}{2k\left({\omega }_n\right)}{\mathcal{F}}_q\left(f\right)\left(n\right){\mathcal{E}}_q\left(x;i{\omega }_n\right).$$

Proof. The result follows from the fact that the family ${\left\{{\mathcal{E}}_q\left(x;i{\omega }_n\right)\right\}}_{n=-\infty }^{\infty }$ is complete and orthogonal in L² ((− 1, 1), w(x)dx) (see [4], [6]). □

Next, we use the q-Fourier series to define the q-translation operators $T_q^x$. Let denote by H _ε , (ε > 0), the space of functions in L²((− 1, 1), w(x)dx) such that

$$\sum _{n=-\infty }^{\infty }\frac{q^{-4\epsilon \left|n\right|}}{2k\left({\omega }_n\right)}|\mathcal{F}\left(f\right)\left(n\right){|}^2<\infty .$$

Definition 1. Let 0 < ε < 1/ 2 and f in H _ε , we put

$$\begin{array}{c}{T}_q^{y}f\left(x\right)=\sum _{n=-\infty }^{\infty }\frac{1}{2k\left({\omega }_n\right)}{\mathcal{F}}_q\left(f\right)\left(n\right){\mathcal{E}}_q\left(x;i{\omega }_n\right){\mathcal{E}}_q\left(y;i{\omega }_n\right),\\ =\sum _{n=-\infty }^{\infty }\frac{1}{2k\left({\omega }_n\right)}{\mathcal{F}}_q\left(f\right)\left(n\right){\mathcal{E}}_q\left(x,y;i{\omega }_n\right).\end{array}$$

The operators $T_q^y$, are called q− translation operators associated to the Askey-Wilson operator.

Remark 1. (1) The q-translation operators are characterized by the formula

$${\mathcal{F}}_q\left(T_q^{y}f\right)\left(n\right)={\mathcal{E}}_q\left(y;i{\omega }_n\right){\mathcal{F}}_q\left(f\right)\left(n\right),n\in \mathbb{Z}.$$

• (2) The Askey-Wilson operator introduced in (7) can be defined on the space H _{1/ 2}via

  $${\mathcal{D}}_{q}f\left(x\right)=\frac{q^{1/4}}{1-q}i\sum _{n=-\infty }^{\infty }\frac{1}{k\left({\omega }_n\right)}{\omega }_n{\mathcal{F}}_q\left(f\right)\left(n\right){\mathcal{E}}_q\left(x;i{\omega }_n\right).$$

Proposition 2. For 0 < ε < 1/ 2, we have

• (1) $T_q^0=id$

• (2) $T_q^y{\mathcal{E}}_q\left(x;i{\omega }_n\right)={\mathcal{E}}_q\left(x;i{\omega }_n\right){\mathcal{E}}_q\left(y;i{\omega }_n\right)$

• (3) $T_q^{y}f\left(x\right)=T_q^{x}f\left(y\right)$

• (4) ${\mathcal{D}}_q{T}_q^{y}f=T_q^y{\mathcal{D}}_{q}f,f\in H_1$,

where id denotes the identity operator.

Proof. The properties (1)-(3) are evident. To prove property (4), let

$$f\left(x\right)=\sum _{n=-\infty }^{\infty }{\mathcal{F}}_q\left(f\right)\left(n\right){\mathcal{E}}_q\left(x;i{\omega }_n\right)\in H_1.$$

From (9) and (10) we have

$$\begin{array}{ll}\hfill {\mathcal{D}}_q{T}_q^{y}f& =\sum _{n=-\infty }^{\infty }\frac{2q^{1/4}}{1-q}i{\omega }_m{\mathcal{F}}_q\left(f\right)\left(n\right){\mathcal{E}}_q\left(x;i{\omega }_n\right){\mathcal{E}}_q\left(x;i{\omega }_n\right)\\ =T_q^y{\mathcal{D}}_{q}f.\end{array}$$

□

Theorem 3. For 0 < ε < 1/ 2 and f in H _ε , the function

$$u\left(x,y\right)=T_q^{y}f\left(x\right),$$

is the unique solution of the system

$$\left\{\begin{array}{c}\hfill {\mathcal{D}}_{q,x}u\left(x,y\right)={\mathcal{D}}_{q,y}u\left(x,y\right),\hfill \\ \hfill u\left(x,0\right)=f\left(x\right).\hfill \end{array}\right.$$

(11)

Proof. It is clear that the function $T_q^{y}f\left(x\right)$ is a solution of the system (11). Applying the q-Fourier transform to each member of the system (11), we obtain for $n\in \mathbb{Z}$,

$$\left\{\begin{array}{c}\hfill {\mathcal{D}}_{q,y}{\mathcal{F}}_q\left(u\left(.;y\right)\right)\left(n\right)=\frac{2i{q}^{1/4}}{1-q}{\omega }_n{\mathcal{F}}_q\left(u\left(.;y\right)\right),\hfill \\ \hfill {\mathcal{F}}_q\left(u\left(.;0\right)\right)\left(n\right)={\mathcal{F}}_q\left(f\right)\left(n\right).\hfill \end{array}\right.$$

Hence

$${\mathcal{F}}_q\left(u\left(.;y\right)\right)\left(n\right)={\mathcal{F}}_q\left(f\right)\left(n\right){\mathcal{E}}_q\left(y;i{\omega }_n\right).$$

So that

$$\begin{array}{ll}\hfill u\left(x,y\right)& =\sum _{n=-\infty }^{\infty }{\mathcal{F}}_q\left(f\right)\left(n\right){\mathcal{E}}_q\left(y;i{\omega }_n\right){\mathcal{E}}_q\left(x;i{\omega }_n\right)\\ =T_q^{y}f\left(x\right).\end{array}$$

□

Proposition 4. Let 0 < ε < 1/ 4, we have

$${\mathcal{E}}_q\left(x;it\right)\in H_{\epsilon }.$$

Proof. From the integral (3.13) in [7], we get

$$\begin{array}{ll}\hfill \mathcal{F}\left({\mathcal{E}}_q\left(.;it\right)\right)\left(n\right)& ={\int }_{-1}^1{\mathcal{E}}_q\left(x;it\right){\mathcal{E}}_q\left(x;-i{\omega }_n\right)w\left(x\right)dx\\ \mathsf{\text{ = Sin}}{\mathsf{\text{c}}}_q\mathsf{\text{(}}t\mathsf{\text{,}}n\mathsf{\text{),}}\end{array}$$

where

$$\mathsf{\text{Sin}}{\mathsf{\text{c}}}_q\left(t,n\right)=\frac{{\left(q^{1/2};q\right)}_{1/2}{\left(-q{\omega }_n^2;q^2\right)}_{\infty }\mathsf{\text{Im}}\left({\left(it,-i{\omega }_n;q^{1/2}\right)}_{\infty }\right)}{{\left(-q{t}^2;q^2\right)}_{\infty }\left({\omega }_n+t\right)\frac{\partial }{\partial t}\mathsf{\text{Im}}\left({\left(it,-i{\omega }_n;q^{1/2}\right)}_{\infty }\right){|}_{t=-{\omega }_n}}.$$

By (9), we have the asymptotic estimates as n → ∞

$$\begin{array}{ll}\hfill {\left(-q{\omega }_n^2;q^2\right)}_{\infty }& ~{\left(-q^{1/2-2n};q^2\right)}_{\infty }\\ ~{\left(-\right)}^n{q}^{-n^2+1/2n}{\left(-q^{1/2};q^2\right)}_n{\left(-q^{3/2};q^2\right)}_{\infty },\end{array}$$

and similarly

$$\mathsf{\text{Im}}{\left(it,-i{\omega }_n;q^{1/2}\right)}_{\infty }~{\left(-1\right)}^n{q}^{-n^2}\mathsf{\text{Im}}{\left(it,i{q}^{1/4};q^{1/2}\right)}_{\infty }.$$

Then from (9) and relation ((3.15), [7]), we obtain as n → ∞

$$\mathsf{\text{Sin}}{\mathsf{\text{c}}}_q\left(t,n\right)~\frac{{\left(q,q^2\right)}_{\infty }^3\mathsf{\text{Im}}{\left(it,i{q}^{1/4};q^{1/2}\right)}_{\infty }}{2q^{1/4}{\left(q,q\right)}_{\infty }^2{\left(q^{-3/2},q^{1/2},q;q^2\right)}_{\infty }}{q}^{n/2},$$

(12)

and

$$\mathsf{\text{Sin}}{\mathsf{\text{c}}}_q\left(t,-n\right)~\frac{{\left(q,q^2\right)}_{\infty }^3\mathsf{\text{Im}}{\left(-it,i{q}^{1/4};q^{1/2}\right)}_{\infty }}{2q^{1/4}{\left(q,q\right)}_{\infty }^2{\left(q^{-3/2},q^{1/2},q,q^2\right)}_{\infty }}{q}^{n/2}.$$

(13)

So that

$$\frac{q^{-4\left|n\right|}}{2k\left({\omega }_n\right)}|\mathcal{F}\left({\mathcal{E}}_q\left(.;it\right)\right)\left(n\right){|}^2~C{q}^{\left(1-4\epsilon \right)\left|n\right|}.$$

Then the following series

$$\sum _{n=-\infty }^{\infty }\frac{q^{-4\epsilon \left|n\right|}}{2k\left({\omega }_n\right)}|\mathcal{F}\left({\mathcal{E}}_q\left(.;it\right)\right)\left(n\right){|}^2,$$

converges iff ε < 1/ 4.

This show for 0 < ε < 1/ 4 we have

$${\mathcal{E}}_q\left(x;it\right)\in H_{\epsilon },$$

and

   □

$${\mathcal{E}}_q\left(x;it\right)=\sum _{n=-\infty }^{\infty }\mathsf{\text{Sin}}{\mathsf{\text{c}}}_q\left(t,n\right){\mathcal{E}}_q\left(x;i{\omega }_n\right).$$

(14)

Proposition 5. For t ≠ −iq^{-1/ 2-n}, n = 0, ± 1, ± 2, . . . we have

$$T_q^y{\mathcal{E}}_q\left(x;it\right)={\mathcal{E}}_q\left(x;it\right){\mathcal{E}}_q\left(y;it\right).$$

Proof. From Proposition 2, we see that the following two functions

$$\left(x,y\right)\to {\mathcal{E}}_q\left(x;it\right){\mathcal{E}}_q\left(y;it\right),$$

and

$$\left(x,y\right)\to T_q^y{\mathcal{E}}_q\left(x;it\right),$$

are solutions of the system

$$\left\{\begin{array}{c}\hfill {\mathcal{D}}_{q,x}u\left(x,y\right)={\mathcal{D}}_{q,y}u\left(x,y\right),\hfill \\ \hfill u\left(x,0\right)={\mathcal{E}}_q\left(x;\alpha \right).\hfill \end{array}\right.$$

The result follows by Theorem 3. □

In the following proposition, we find the invariance of the measure w(x)dx over (− 1, 1) by the q-translation operators.

Proposition 6. Let 0 < ε < 1/ 2 and f ∈ H _ε . Then

$${\int }_{-1}^1{T}_q^{y}f\left(x\right)w\left(x\right)dx={\int }_{-1}^{1}f\left(x\right)w\left(x\right)dx.$$

Proof. We have

$${\int }_{-1}^1{T}_q^{y}f\left(x\right)w\left(x\right)dx={\int }_{-1}^1\sum _{n=-\infty }^{\infty }\frac{1}{2k\left({\omega }_n\right)}\mathcal{F}\left(f\right)\left(n\right){\mathcal{E}}_q\left(x;i{\omega }_n\right){\mathcal{E}}_q\left(y;i{\omega }_n\right)w\left(x\right)dx.$$

Then, we get after interchangement of integral and sum

$${\int }_{-1}^1{T}_q^{y}f\left(x\right)w\left(x\right)dx=\mathcal{F}\left(f\right)\left(0\right)={\int }_{-1}^{1}f\left(x\right)w\left(x\right)dx.$$

(15)

To justify the interchangement of integral and summation, we put

$$\sum _{n=-\infty }^{\infty }{\int }_{-1}^1\left|\frac{1}{2k\left({\omega }_n\right)}\mathcal{F}\left(f\right)\left(n\right){\mathcal{E}}_q\left(x;i{\omega }_n\right){\mathcal{E}}_q\left(y;i{\omega }_n\right)\right|w\left(x\right)dx=\sum _{-\infty }^{\infty }{c}_n.$$

(16)

Let 0 < η < 1/ 4 and δ > 0 such that 2η + δ < 1/ 2, by (9) and (10), we have

$$\begin{array}{ll}\hfill c_n& ~\frac{1}{2k\left({\omega }_n\right)}\left|\mathcal{F}\left(f\right)\left(n\right)\right|q^{-4\eta \left|n\right|},\\ ~\frac{1}{2k\left({\omega }_n\right)}\left|\mathcal{F}\left(f\right)\left(n\right)\right|q^{-2\left(2\eta +\delta \right)\left|n\right|}{q}^{2\delta \left|n\right|}.\end{array}$$

The convergence of the series ${\sum }_{-\infty }^{\infty }{c}_n$ follows from the Cauchy inequality and the fact that f ∈ H _ε .

□

In the following proposition, we show that the q-translation are self-adjoint operators.

Proposition 7. Let f and g ∈ H _ε where 0 < ε < 1/ 2. Then

$${\int }_{-1}^1{T}_q^{x}f\left(-y\right)g\left(y\right)w\left(y\right)dy={\int }_{-1}^{1}f\left(y\right)T_q^{x}g\left(-y\right)w\left(y\right)dy.$$

Ismail [1] proved that the q-exponential function ${\mathcal{E}}_q\left(x;\alpha \right)$ is the only solution of the functional equation

$$f\left(x\oplus y\right)=f\left(x\right)f\left(y\right),$$

(17)

where f (x) has the expansion $f\left(x\right)=\sum _{n=0}^{\infty }\frac{f_n}{{\left(q;q\right)}_n}{g}_n\left(x\right)$, which converges uniformly on compact subsets of a domain Ω.

For f ∈ L² ((− 1, 1), w(x)dx), we put

$$\hat{f}\left(t\right)={\int }_{-1}^{1}f\left(x\right){\mathcal{E}}_q\left(x;-it\right)w\left(x\right)dx.$$

Proposition 8. Let f ∈ L²((− 1, 1) w(x)dx), then the function

$$F\left(t\right)={\left(-q{t}^2;q^2\right)}_{\infty }\hat{f}\left(t\right),$$

is an entire function such that

$$\underset{r\to \infty }{\text{lim}}\frac{\text{ln}\left(M\left(r,F\right)\right)}{{\text{ln}}^{2}r}\le \frac{1}{\text{ln}q-1}.$$

Furthermore, the function F is of order 0 and has infinitely many zeros.

Proof. The function f is in L²((− 1, 1) w(x)dx), then

$${\int }_{-1}^1\left|f\left(x\right)\right|w\left(x\right)dx<\infty .$$

From (2) we have the following estimate

$$\left|H_n\left(x|q\right)\right|\le H_n\left(1|q\right),for\left|x\right|<1,$$

and by (3) we can write for all t ∈ $ℂ$

$$\left|F\left(t\right)\right|\le {\int }_{-1}^1\left|f\left(x\right)\right|w\left(x\right)dx{\left(-q|t{|}^2;q^2\right)}_{\infty }{\mathcal{E}}_q\left(1;\left|t\right|\right).$$

On the other hand

$${\left(-q|t{|}^2;q^2\right)}_{\infty }{\mathcal{E}}_q\left(1;\left|t\right|\right)=\sum _{n=0}^{\infty }\frac{{\alpha }^n}{{\left(q;q\right)}_n}{q}^{n^2/4}{H}_n\left(1|q\right).$$

The result follows by a similar proof as in Lemma 14.1.4 and Corollary 14.1.5 in [3]. □

Proposition 9. We have

$$\widehat{T_q^{y}f}\left(t\right)={\mathcal{E}}_q\left(y;it\right)\hat{f}\left(t\right).$$

Proof. It is easy to see that the function w(x) is even and the q-exponential ${\mathcal{E}}_q\left(x;it\right)$ satisfies

$${\mathcal{E}}_q\left(-x;it\right)={\mathcal{E}}_q\left(x;-it\right).$$

Then from Propositions 4 and 7, we get

$$\begin{array}{ll}\hfill \widehat{T_q^{y}f}\left(t\right)& ={\int }_{-1}^1{T}_q^{y}f\left(x\right){\mathcal{E}}_q\left(-x;it\right)w\left(x\right)dx\\ ={\int }_{-1}^{1}f\left(x\right)T_q^y{\mathcal{E}}_q\left(-x;it\right)w\left(x\right)dx\\ ={\mathcal{E}}_q\left(y;it\right){\int }_{-1}^{1}f\left(x\right){\mathcal{E}}_q\left(-x;it\right)w\left(x\right)dx\\ ={\mathcal{E}}_q\left(y;it\right)\hat{f}\left(t\right).\end{array}$$

□

In the following Proposition, we show that the q-translation operator $T_q^x$ coincides with the q-Translation $E_q^x$ defined by Ismail on the set of q-Hermite polynomials (2).

Proposition 10. For n = 0, 1, 2, . . . , we have

$$T_q^x{H}_n\left(y|q\right)=E_q^x{H}_n\left(y|q\right).$$

(18)

Proof. By Proposition 5, we get

$${\left(-q{t}^2;q^2\right)}_{\infty }{T}_q^x\mathcal{E}\left(y;it\right)={\left(-q{t}^2;q^2\right)}_{\infty }\mathcal{E}\left(x;it\right)\mathcal{E}\left(y;it\right).$$

Then the formula ([3, 14.6.7]) and (3) lead to

$$\sum _{n=0}^{\infty }\frac{{\left(it\right)}^n{q}^{n^2/4}}{{\left(q;q\right)}_n}{T}_q^y{H}_n\left(x|q\right)=\sum _{n=0}^{\infty }\frac{g_n\left(y\right)}{{\left(q;q\right)}_n}{\left(it\right)}^n\sum _{m=0}^{\infty }\frac{{\left(it\right)}^m{q}^{m^2/4}}{{\left(q;q\right)}_m}{H}_m\left(x|q\right).$$

Hence,

$$T_q^y{H}_n\left(x|q\right)=\sum _{m=0}^n\frac{{\left(q;q\right)}_n}{{\left(q;q\right)}_m{\left(q;q\right)}_{n-m}}{H}_m\left(x|q\right)g_{n-m}\left(y\right)q^{\left(m^2-n^2\right)/4}.$$

□

Theorem 11. Let f be a function in L²((− 1, 1), w(x)dx) satisfying the following functional equation

$$T_q^{y}f\left(x\right)=f\left(x\right)f\left(y\right),$$

and we denote by Z _f the set of zeros of

$$F\left(t\right)={\left(-q{t}^2;q^2\right)}_{\infty }\hat{f}\left(t\right).$$

Then f is a function of two variables x and t equal to the q-exponential function${\mathcal{E}}_q\left(x;it\right)$, for |x| < 1 and t ∈ $ℂ$ − Z _f

Proof. Let f be a function in L² ((− 1, 1), w(x)dx) satisfying

$$T_q^{y}f\left(x\right)=f\left(x\right)f\left(y\right).$$

By Proposition 9, we have

$${\mathcal{E}}_q\left(x;it\right)\hat{f}\left(t\right)=f\left(x\right)\hat{f}\left(t\right).$$

Then for all complex numbers t such that $\hat{f}\left(t\right)\ne 0$, we have

$$f\left(x\right):=f\left(x,t\right)={\mathcal{E}}_q\left(x;it\right),$$

and f is a function of two variables x and t. □

q-Gauss Weierstrass transform

We conclude this study by an application of the q-translation operators. We consider the q-analogue of the Gauss Weierstrass transform by (see [3]).

$$F_W\left(f\right)\left(y\right)=\frac{{\left(q;q\right)}_{\infty }}{2\pi }{\int }_0^{\pi }{T}_q^{y}f\left(x\right)W\left(x\right)dx,$$

(19)

where

$$W\left(x\right)={\left(e^{2i\theta },e^{-2i\theta };q\right)}_{\infty }.$$

In [1], the author proved that (19) can be inverted by the Askey-Wilson operator

$$f\left(y\right)={\left(\frac{1}{4}{q}^{1/2}{\left(1-q\right)}^2{\mathcal{D}}_q^2;q^2\right)}_{\infty }{F}_W\left(f\right)\left(y\right).$$

(20)

where f is a polynomial.

In the following theorem we prove that the inversion formula (20) is still valid in the space

$$H_{\infty }=\bigcap _{\epsilon >0}{H}_{\epsilon }.$$

Theorem 12. The q-Gauss Weierstrass transform has the inversion formula

$$f\left(y\right)={\left(\frac{1}{4}{q}^{1/2}{\left(1-q\right)}^2{\mathcal{D}}_q^2;q^2\right)}_{\infty }{F}_W\left(y\right),f\in H_{\infty }.$$

Proof. Let f ∈ H_∞, then by the formula

$$\frac{1}{{\left(q{t}^2;q^2\right)}_{\infty }}=\frac{{\left(q;q\right)}_{\infty }}{2\pi }{\int }_0^{\pi }{\mathcal{E}}_q\left(x;t\right)W\left(x\right)dx,$$

we have

$$\begin{array}{ll}\hfill F_W\left(y\right)& =\frac{{\left(q;q\right)}_{\infty }}{2\pi }\sum _{n=-\infty }^{\infty }\frac{1}{2k\left({\omega }_n\right)}{\mathcal{F}}_q\left(f\right)\left(n\right){\mathcal{E}}_q\left(y;i{\omega }_n\right){\int }_{-1}^1{\mathcal{E}}_q\left(x;i{\omega }_n\right)W\left(x\right)dx\\ =\sum _{n=-\infty }^{\infty }\frac{1}{2k\left({\omega }_n\right)}\frac{1}{{\left(-q{\omega }_n^2;q^2\right)}_{\infty }}{\mathcal{F}}_q\left(f\right)\left(n\right){\mathcal{E}}_q\left(y;i{\omega }_n\right).\end{array}$$

So that

$$\begin{array}{c}{\left(\frac{1}{4}{q}^{1/2}{\left(1-q\right)}^2{\mathcal{D}}_q^2;q^2\right)}_{\infty }{F}_W\left(x\right)\\ =\sum _{n=-\infty }^{\infty }\frac{1}{2k\left({\omega }_n\right)}{\mathcal{F}}_q\left(f\right)\left(n\right){\mathcal{E}}_q\left(y;i{\omega }_n\right)\\ =f\left(y\right)\end{array}$$

□

Another q-analogue of Gauss Weierstrass transform can be defined by

$$F_{\gamma }\left(y\right)=\frac{{\left(q,{\gamma }^2;q\right)}_{\infty }}{2\pi {\left(\gamma ,q{\gamma }^2;q\right)}_{\infty }}{\int }_{-1}^1{T}_q^{y}f\left(x\right)w\left(x|\gamma ,q\right)dx,$$

where

$$w\left(\text{cos}\theta |\gamma ,q\right)=\frac{{\left(e^{2i\theta },e^{-2i\theta };q\right)}_{\infty }}{{\left(\gamma e^{2i\theta },\gamma e^{-2i\theta };q\right)}_{\infty }}.$$

In a similar way as in Theorem 12, we can prove the following inversion formula for the transform F _γ .

Theorem 13. The transform F_γ has the inversion formula

$$f\left(y\right)={\phi }_{\gamma }\left(\frac{1}{4}{q}^{1/2}{\left(1-q\right)}^2{\mathcal{D}}_q^2\right)F_{\gamma }\left(y\right),f\in H_{\infty },$$

where

$${\phi }_{\gamma }(\alpha )=\frac{1}{{}_0{\varphi }_1(-;q\gamma ;q;q\gamma {\alpha }^2)}.$$