Generalized q-Bernoulli Polynomials Generated by Jackson q-Bessel Functions

Abstract

In this paper, we introduce the polynomials B^{(k)}_{n,alpha }(x;q) generated by a function including Jackson q-Bessel functions J^{(k)}_{alpha }(x;q) (k=1,2,3),,alpha >-1. The cases alpha =pm frac{1}{2} are the q-analogs of Bernoulli and Euler^{,}s polynomials introduced by Ismail and Mansour for (k=1,2), Mansour and Al-Towalib for (k=3). We study the main properties of these polynomials, their large n degree asymptotics and give their connection coefficients with the q-Laguerre polynomials and little q-Legendre polynomials.