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

TL;DR

This paper introduces a new family of generalized $q$-Bernoulli polynomials generated via Jackson $q$-Bessel functions, exploring their properties, asymptotics, and connections to other $q$-orthogonal polynomials.

Contribution

It defines a new class of $q$-Bernoulli polynomials linked to Jackson $q$-Bessel functions and analyzes their properties and relations to existing $q$-polynomials.

Findings

Derived properties and asymptotic behavior of the polynomials

Established connection coefficients with $q$-Laguerre and little $q$-Legendre polynomials

Extended the framework of $q$-Bernoulli and Euler polynomials

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.