Carlitz q-Bernoulli numbers and continued fractions

TL;DR

This paper explores q-analogues of Bernoulli numbers introduced by Carlitz, providing new representations, determinant factorizations, and continued fractions, some of which extend classical results to the q-analogue setting.

Contribution

It introduces orthogonal polynomial representations and continued fractions for Carlitz q-Bernoulli numbers, extending classical Bernoulli number results to the q-analogue context.

Findings

Representation of q-Bernoulli numbers as moments of orthogonal polynomials

Factorizations of Hankel determinants of q-Bernoulli numbers

Continued fractions for their generating series

Abstract

Carlitz has introduced q-analogues of the Bernoulli numbers around 1950. We obtain a representation of these q-Bernoulli numbers (and some shifted version) as moments of some orthogonal polynomials. This also gives factorisations of Hankel determinants of q-Bernoulli numbers, and continued fractions for their generating series. Some of these results are q-analogues of known results for Bernoulli numbers, but some are specific to the q-Bernoulli setting.