Generalized $q$-Bernoulli polynomials generated by Jackson $q$-Bessel functions

Abstract

In this paper, we introduce the polynomials $B_{n,\alpha }^{(k)}(x;q)$ generated by a function including Jackson $q$-Bessel functions $J_{\alpha }^{(k)}(x;q)$ $(k=1,2,3),\alpha >-1$. 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.