A determinant approach for generalized $q$-Bernoulli polynomials and asymptotic results

TL;DR

This paper develops determinant representations and asymptotic analysis for three families of generalized $q$-Bernoulli polynomials linked to Jackson $q$-Bessel functions, including expansion theorems and illustrative examples.

Contribution

It introduces determinant formulas and asymptotic results for these polynomial families, expanding understanding of their structure and applications.

Findings

Determinant representations for each polynomial family

Asymptotic formulas for large degree $n$

Expansion theorems for specific entire functions

Abstract

In earlier work, we introduced three families of polynomials where the generating function of each set includes one of the three Jackson $q$-analogs of the Bessel function. This paper gives determinant representation for each family, their large $n $ asymptotics, and two expansion theorems for specific classes of entire functions. We include two examples.