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.
Categories: math.NT