Relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index

Abstract

Poly-Bernoulli numbers $B_n^{(k)}\in\mathbb{Q}$\,($n \geq 0$,\,$k \in \mathbb{Z}$) are defined by Kaneko in 1997. Multi-Poly-Bernoulli numbers\,$B_n^{(k_1,k_2,\ldots, k_r)}$, defined by using multiple polylogarithms, are generations of Kaneko's Poly-Bernoulli numbers\,$B_n^{(k)}$. We researched relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index in particular. In section 2, we introduce a identity for Multi-Poly-Bernoulli numbers of negative index which was proved by Kamano. In section 3, as main results, we introduce some relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index in particular.