Expansion and asymptotic in terms of basic Bessel functions

This work aims to study the expansion and asymptotic for solutions of q  -difference equations in terms of the basic Bessel functions, namely Jα(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].