Explicit formula for multi-indexed poly-Bernoulli numbers

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 ${\mathbb{B}}_{m_1,\dots ,m_r}^{(k_1,\dots ,k_r)}$ using the multiple polylogarithm and reached their duality properties via an associated $\hat{I}\cdot$-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.