A polynomial approach to Carlitz's $q$-Bernoulli numbers

TL;DR

This paper introduces a new polynomial sequence using Jackson integrals that provides a novel polynomial framework for Carlitz's $q$-Bernoulli numbers, connecting classical and $q$-analogues.

Contribution

It presents a new polynomial sequence defined via Jackson integral that serves as a $q$-analogue of Bernoulli polynomials, linking to Carlitz's $q$-Bernoulli numbers.

Findings

The numbers $B_{n , q}(0)$ are exactly Carlitz $q$-Bernoulli numbers.

The polynomials $B_{n , q}(X)$ are genuine $q$-analogues of classical Bernoulli polynomials.

The Jackson integral formulation offers new insights into properties of $q$-Bernoulli numbers.

Abstract

This paper investigates $q$-analogues of the classical Bernoulli polynomials and numbers. We introduce a new polynomial sequence ${\left(B_{n , q}(X)\right)}_{n \in \mathbb{N}_0}$, defined via the Jackson integral, and explore its connections with Carlitz's $q$-Bernoulli polynomials and numbers. Specifically, we prove that the numbers $B_{n , q}(0)$ are exactly the Carlitz $q$-Bernoulli numbers and that the polynomials $B_{n , q}(X)$ are genuine $q$-analogues of the classical Bernoulli polynomials. This approach leverages the Jackson integral to reformulate Carlitz's $q$-Bernoulli numbers in terms of classical polynomial structures, offering new insights into their properties.