2025/07/27 by Mouzaia, Mohamed, Farhi, Bakir
#FOS: Mathematics #Number Theory (math.NT) #Primary 05A30 #Secondary 11B68
paper · doi:10.48550/arxiv.2507.20384
This paper investigates q-analogues of the classical Bernoulli polynomials and numbers. We introduce a new polynomial sequence (Bn , q(X))n ∈ ℕ0, defined via the Jackson integral, and explore its connections with Carlitz's q-Bernoulli polynomials and numbers. Specifically, we prove that the numbers Bn , q(0) are exactly the Carlitz q-Bernoulli numbers and that the polynomials Bn , 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.