Multi-indexed poly-Bernoulli numbers

TL;DR

This paper extends the understanding of multi-indexed poly-Bernoulli numbers by deriving explicit formulas, periodicity properties, and relations among various variants, building on known properties of poly-Bernoulli numbers.

Contribution

It provides new explicit formulas and periodicity results for double-indexed poly-Bernoulli numbers and explores relations with star-version variants, advancing the theory of poly-Bernoulli numbers.

Findings

Explicit formula using Stirling numbers of the second kind for double-indexed poly-Bernoulli numbers

Periodicity property for negative upper-index cases

Relations between multi-indexed and star-version poly-Bernoulli numbers

Abstract

As properties of poly-Bernoulli numbers, a number of formulas such as the duality formula, explicit formula using the Stirling numbers of the second kind and periodicity for negative upper-index have been established. For the multi-indexed poly-Bernoulli numbers generalized by Kaneko-Tsumura, among such properties only the duality formula was obtained. In this paper, we restrict the double-indexed poly-Bernoulli numbers and show the explicit formula using the Stirling numbers of the second kind and periodicity for negative upper-index for them. Further, we define the variant of multiple-indexed poly-Bernoulli numbers using the star-version of multiple-indexed logarithms and obtain the relation between this kind of double and triple-indexed poly-Bernoulli numbers with multi-indexed poly-Bernoulli numbers ahead.