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On the q-translation associated with the Askey-Wilson operator
Advances in Difference Equations volume 2012, Article number: 87 (2012)
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.
The concept of the q-translation operators introduced by Ismail  was defined in polynomials through their action on the continuous q-Hermite polynomials H m (x | q) as follows
In others words
In (, (2.21)), the author proved the following product formula for the q-exponential function
Furthermore, if y and z are two complex variables, then we have 
In  problem 24.5.6, Ismail proposed the extension of the action of to measurable functions and proving that the only measurable functional solution of the q-analogue of the Cauchy functional equation
is the q-exponential function. . The purpose of this paper is to define a new q-translation operator 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 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 . In Section "Preliminaries", we define and study the q-translation operator . Also we solve the following problem
As a consequence of (5) we solve the basic analogue of the Cauchy functional equation
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".
Let 0 < q < 1 and a ∈ , the q-shift factorial is defined by (see )
Given a function f (x) with x = cos θ, f (x) can be viewed as a function of eiθ.
The Askey-Wilson-divided difference operator is defined by
The q-exponential function is given by 
The q-exponential function is a solution of the q-difference equation of first-order 
then, we have (see )
and used transformation formulas to continue them analytically to entire functions in the variable ω. Bustoz and Suslov  have established the following orthogonality relations
ω0 = 0, ω1 < ω2 < ..., are zeros of the q-sine function S q ((q1/ 4+ q-1/ 4)/2); ω) and for n = 1, 2, . . . , ω -n = −ω n .
From , we have the following asymptotic estimates as n → ∞
and for 0 < ε < 1/ 2 and |x| ≤ 1 < (qε + q-ε)/ 2, we have
where C = 1/(−q1/ 2, q; q)∞(qε, q1/ 2-ε; q1/ 2)∞.
We define the q-Fourier transform as
Theorem 1. The transform is an isomorphism from L2((− 1, 1), w(x)dx) into l2 (k (ω n )) and its inverse is given by
Next, we use the q-Fourier series to define the q-translation operators . Let denote by H ε , (ε > 0), the space of functions in L2((− 1, 1), w(x)dx) such that
Definition 1. Let 0 < ε < 1/ 2 and f in H ε , we put
The operators , are called q− translation operators associated to the Askey-Wilson operator.
Remark 1. (1) The q-translation operators are characterized by the formula
The Askey-Wilson operator introduced in (7) can be defined on the space H 1/ 2via
Proposition 2. For 0 < ε < 1/ 2, we have
where id denotes the identity operator.
Proof. The properties (1)-(3) are evident. To prove property (4), let
From (9) and (10) we have
Theorem 3. For 0 < ε < 1/ 2 and f in H ε , the function
is the unique solution of the system
Proof. It is clear that the function is a solution of the system (11). Applying the q-Fourier transform to each member of the system (11), we obtain for ,
Proposition 4. Let 0 < ε < 1/ 4, we have
Proof. From the integral (3.13) in , we get
By (9), we have the asymptotic estimates as n → ∞
Then from (9) and relation ((3.15), ), we obtain as n → ∞
Then the following series
converges iff ε < 1/ 4.
This show for 0 < ε < 1/ 4 we have
Proposition 5. For t ≠ −iq-1/ 2-n, n = 0, ± 1, ± 2, . . . we have
Proof. From Proposition 2, we see that the following two functions
are solutions of the system
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
Proof. We have
Then, we get after interchangement of integral and sum
To justify the interchangement of integral and summation, we put
Let 0 < η < 1/ 4 and δ > 0 such that 2η + δ < 1/ 2, by (9) and (10), we have
The convergence of the series 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
Ismail  proved that the q-exponential function is the only solution of the functional equation
where f (x) has the expansion , which converges uniformly on compact subsets of a domain Ω.
For f ∈ L2 ((− 1, 1), w(x)dx), we put
Proposition 8. Let f ∈ L2((− 1, 1) w(x)dx), then the function
is an entire function such that
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
From (2) we have the following estimate
and by (3) we can write for all t ∈
On the other hand
The result follows by a similar proof as in Lemma 14.1.4 and Corollary 14.1.5 in . □
Proposition 9. We have
Proof. It is easy to see that the function w(x) is even and the q-exponential satisfies
Then from Propositions 4 and 7, we get
In the following Proposition, we show that the q-translation operator coincides with the q-Translation defined by Ismail on the set of q-Hermite polynomials (2).
Proposition 10. For n = 0, 1, 2, . . . , we have
Proof. By Proposition 5, we get
Then the formula ([3, 14.6.7]) and (3) lead to
Theorem 11. Let f be a function in L2((− 1, 1), w(x)dx) satisfying the following functional equation
and we denote by Z f the set of zeros of
Then f is a function of two variables x and t equal to the q-exponential function, for |x| < 1 and t ∈ − Z f
Proof. Let f be a function in L2 ((− 1, 1), w(x)dx) satisfying
By Proposition 9, we have
Then for all complex numbers t such that , we have
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 ).
In , the author proved that (19) can be inverted by the Askey-Wilson operator
where f is a polynomial.
In the following theorem we prove that the inversion formula (20) is still valid in the space
Theorem 12. The q-Gauss Weierstrass transform has the inversion formula
Proof. Let f ∈ H∞, then by the formula
Another q-analogue of Gauss Weierstrass transform can be defined by
In a similar way as in Theorem 12, we can prove the following inversion formula for the transform F γ .
Theorem 13. The transform Fγ has the inversion formula
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The author thank the referees for her careful reading of the manuscript and for helpful suggestions. This research was supported by the NPST Program of King Saud University, project number 10-MAT1293-02.
The authors declare that they have no competing interests.
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Bouzaffour, F. On the q-translation associated with the Askey-Wilson operator. Adv Differ Equ 2012, 87 (2012). https://doi.org/10.1186/1687-1847-2012-87
- basic orthogonal polynomials and functions
- basic hypergeometric integrals