Fethi Bouzeffour

Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

Nonsymmetric Askey–Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl–Cherednik operator of which they are eigenfunctions, is represented as a 2 × 2 matrix-valued operator. As a new result made possible by this approach we obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as limits both of the Askey–Wilson and of the little q-Jacobi case.

Interpolation of entire functions, product formula for basic sine function

Abstract We solve the problem of constructing entire functions where ln M (r; f) grows like ln2 r from their values at q−n, for 0 < q < 1. As application we give a product formula for the basic sine function.

A

In this paper we derive a q-analogue of the sampling theorem for Jacobi functions. We also establish a product formula for the nonterminating version of the q-Jacobi polynomials. The proof uses recent results in the theory of q-orthogonal polynomials and basic hypergeometric functions.

q

Abstract A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Askey-Wilson functions is proved. Applications to finite continuous Askey-Wilson transform are given.

-sampling theorem and product formula for continuous

We define an analogue of the Paley-Wiener space in the context of the Askey-Wilson function transform, compute explicitly its reproducing kernel and prove that the growth of functions in this space of entire functions is of order two and type lnq 1 , providing a Paley-Wiener Theorem for the Askey-Wilson transform. Up to a change of scale, this growth is related to the refined concepts of exponential order and growth proposed by J. P. Ramis. The Paley-Wiener theorem is proved by combining a sampling theorem with a result on interpolation of entire functions due to M. E. H. Ismail and D. Stanton.

q

In this work, we establish a Babenko–Beckner-type inequality for the Dunkl transform of a radial function and Dunkl transform associated to a finite reflection group generated by the sign changes. As applications, we establish a new version of Young’s-type inequality and uncertainty relation for Renyi entropy.

-Jacobi functions

In this paper we express the matrix coefficients of the Fock representation of a q-oscillator algebra in terms of the d-orthogonal Al-Salam Carlitz polynomials. Also, we derive a generating functions, recurrence relations and q-difference equations for these d-orthogonal polynomials.

A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Askey-Wilson functions

Abstract This work aims to study the expansion and asymptotic for solutions of q-difference equations in terms of the basic Bessel functions, namely J α ( 2 ) ( x ; q ) . For this purpose, we will show that the constructive method introduced by Olver [F.W.J. Olver, Asymptotics and Special Functions, Academic Press, Inc., 1974] and exploited early by Fitouhi et al. [H. Chebli, A. Fitouhi, M.M. Hamza, Expansion in series of Bessel functions and transmutations for perturbed Bessel operator, J. Math. Anal. Appl. 181 (3) (1994); A. Fitouhi, M.M. Hamza, Uniform expansion for eigenfunction of singular second order differential operator, SIAM J. Math. Anal. 21 (6) (1990); A. Fitouhi, M.M. Hamza, Expansion in series of Laguerre functions for solution of perturbed Laguerre equations, Contemp. Math. 183 (1995); A. Fitouhi, J. El Kamel, Expansion in series of Gegenbauer polynomials, Int. Trans. Sp. Funct. 5 (3–4) (1997) 213–226] can be extended in this context. As application new expansions of some basic special functions are established in particular these given by Ismail [Y. Chen, M.E.H. Ismail, K.A. Muttalib, Asymptotics of basic Bessel functions and q-Laguerre polynomilas, J. Comput. Appl. Math. 54 (1994) 263–272; M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005].

A Paley-Wiener theorem for the Askey-Wilson function transform

We discuss weak convergence of the number of busy servers in a G/G/∞ queue in the J 1 -topology on the Skorokhod space. We prove two functional limit theorems with random and nonrandom centering, thereby solving two open problems stated in Mikosch and Resnick (2006). A new integral representation for the limit Gaussian process is given.

On the norm of the Lp-Dunkl transform

The aim of this paper is to prove Heisenberg-Pauli-Weyl inequality for a fractional power of the Dunkl transform on the real line for which there is an index law and a Plancherel theorem.