Expansion and asymptotic in terms of basic Bessel functions Ahmed Fitouhi a, * , Fethi Bouzeffour b , Wafa Binous c a Faculte ´ des Sciences de Tunis, 1060 Tunis, Tunisia b Institut Pre ´paratoire aux E ´ tudes d’Inge ´nieur de Bizerte, 8000 Bizerte, Tunisia c Institut Pre ´paratoire aux E ´ tudes d’Inge ´nieur de Tunis, Tunisia 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 ð2Þ a ðx; qÞ. 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. Fit- ouhi, 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 dif- ferential 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]. Ó 2006 Published by Elsevier Inc. Keywords: Asymptotic; q-Difference operators; q-Special functions 1. Introduction In [1], Chen et al., established an asymptotic expansion for q-Bessel functions and q-Laguerre polynomials. These expansions are justified by the study of an exactly solvable random transfer matrix model for disordered electronic systems. Many other authors [1,4,5,9] gave asymptotic behavior for the q-special functions via clas- sical methods of analysis. Since in the last decades the Quantum Calculus taken a big progress especially in physical and in the q-partition theory and since the asymptotic expansions play a central role in many fields, we try to contribute in this way by giving some expansions in series and asymptotic expansions of solutions of perturbed q-difference equation of q-Bessel type. We use the same technic used by the first author which is inspired by Olver book’s [10] and based only over the properties of the principal part of the q-difference equa- tion and recursive relations of its eigenfunctions. 0096-3003/$ - see front matter Ó 2006 Published by Elsevier Inc. doi:10.1016/j.amc.2006.11.044 * Corresponding author. E-mail address: Ahmed.Fitouhi@fst.rnu.tn (A. Fitouhi). Applied Mathematics and Computation 188 (2007) 2034–2044 www.elsevier.com/locate/amc