Multi-indexed poly-Bernoulli numbers

As properties of poly-Bernoulli numbers, a number of formulas such as the duality formula, an explicit formula using the Stirling numbers of the second kind, and a 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 has been obtained. In this paper, we restrict the double-indexed poly-Bernoulli numbers and show an explicit formula using the Stirling numbers of the second kind and a periodicity for the negative upper-index for this restriction.Further, we define a variant of multi-indexed poly-Bernoulli numbers using the star-version of multiple polylogarithms and write this kind of double or triple-indexed poly-Bernoulli numbers in terms of original multi-indexed poly-Bernoulli numbers. MSC Classification: 11M68