A polynomial approach to Carlitz's $q$-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.