Ahmed Fitouhi

Asymptotic Approximations in Quantum Calculus

Abstract This paper aims to study the asymptotic approximation of some functions defined by the q-Jackson integrals, for a fix q ∈]0, 1[. For this purpose, we shall attempt to extend the classical methods by giving their q-analogues. In particular, a q-analogue of the Watson’s lemma is discussed and new asymptotic expansions of the q−j α Bessel function and of the q-complementary error function are established.

On Hardy’s inequality for symmetric integral transforms and analogous

Abstract In this paper, we give a generalization of Hardy’s theorem for a class of symmetric transforms in both classical and Quantum calculus. As applications, we give q-analogues of Hardy’s theorem for the q-Fourier-cosine, q-Bessel Fourier and q-Dunkl transforms.

Wavelet Transforms in Quantum Calculus

Abstract This paper aims to study the q-wavelets and the q-wavelet transforms, using only the q-Jackson integrals and the q-cosine Fourier transform, for a fix q ∈]0, 1[. For this purpose, we shall attempt to extend the classical theory by giving their q-analogues.

Some inequalities for the q-beta and the q-gamma functions via some q-integral inequalities

Some new inequalities for the q-gamma, the q-beta and the q-analogue of the Psi functions are established via some q-integral inequalities.

Tauberian Theorems In Quantum Calculus

Abstract In this paper we attempt to establish some tauberian theorems in quantum calculus. This constitutes the beginning of the study of the q-analogue of analytic theory of numbers which is the aim of a forthcoming paper.

q-Bessel positive definite and D q -completely monotonic functions

In this paper we characterize the subspace of Lq,1,v of function which are the q-Bessel Fourier transform of positive functions in Lq,1,v. As application we give a q-version of the Bochner’s theorem.

Expansion and asymptotic in terms of basic Bessel 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].

Inequalities related to the third Jackson q -Bessel function of order zero

In Bettaibi and Bouzeffour (J. Math. Anal. Appl. 342:1203-1219, 2008), some properties of the third Jackson q-Bessel function of order zero were established. This paper is devoted to studying the q-convolution product by using a q-integral representation of the related q-translation.The central part of this work is first to study the related q-heat semi-group and its hypercontractivity and second to specify the q-analogue of the Wiener algebra.

On harmonic analysis related with the generalized Dunkl operator

This paper deals with a new singular differential-difference operator Y t, A on the real line including, as a particular case, the Dunkl, Dunkl–Heckman and Dunkl–Cherednik operators. We establish some results of Harmonic analysis related to this operator, such that a product formula for the related eigenfunction G λ is given as integral with an explicit kernel. This product formula is an important tool to define the generalized translation operator which is used to set up a convolution structure. Next, we establish an inversion formula and prove a generalized Plancherel theorem for this operator. As a direct application, we give a maximum principle of the operators and we solve the heat equation associated with the generalized Dunkl operator on the real line.

On q-Gamma and q-Bessel Functions

In this paper, we present some characterizations of the q-Gamma and the properties of the q-Bessel functions.