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

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 = +/- 1/2 are the q-analogs of Bernoulli and Eulers 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.

Automated summary

In this paper, we introduce a class of 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. We study the main properties of these polynomials, their large n degree asymptotics, and provide their connection coefficients with the q-Laguerre polynomials and little q-Legendre polynomials.