Explicit formula for multi-indexed poly-Bernoulli numbers

TL;DR

This paper derives an explicit formula for multi-indexed poly-Bernoulli numbers, extending previous results and providing an alternative proof of their duality properties, thereby advancing the understanding of these special number sequences.

Contribution

The paper introduces a general explicit formula for multi-indexed poly-Bernoulli numbers, expanding upon prior specific cases and offering a new proof of their duality relations.

Findings

Explicit formula for general multi-indexed poly-Bernoulli numbers

Extension of duality properties to multi-indexed cases

Alternative proof of duality using the new formula

Abstract

The classical Bernoulli numbers $B_m$ can be expressed using Stirling numbers of the second kind, and M. Kaneko extended this framework by defining poly-Bernoulli numbers ${\mathbb B}_m^{(k)}$, for which explicit formulas using the Stirling numbers of the second kind and duality relations were obtained. Later, Kaneko and H. Tsumura introduced multi-indexed poly-Bernoulli numbers ${\mathbb B}_{m_1, \ldots, m_r}^{(k_1, \ldots, k_r)}$ using the multiple polylogarithm and reached their duality properties via an associated $\eta$-function. Explicit formulas for double-indexed poly-Bernoulli numbers ${\mathbb B}_{m_1, m_2}^{(k_1, k_2)}$ were obtained by Y. Baba, M. Nakasuji, and M. Sakata. In this article, we extend these results to general multi-indexed poly-Bernoulli numbers and use it to give an alternative proof of the duality of multi-indexed poly-Bernoulli numbers.