  As properties of poly-Bernoulli numbers, a number of formulas such as the
duality formula, explicit formula using the Stirling numbers of the second kind
and 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 was obtained. In this paper, we restrict
the double-indexed poly-Bernoulli numbers and show the explicit formula using
the Stirling numbers of the second kind and periodicity for negative
upper-index for them. Further, we define the variant of multiple-indexed
poly-Bernoulli numbers using the star-version of multiple-indexed logarithms
and obtain the relation between this kind of double and triple-indexed
poly-Bernoulli numbers with multi-indexed poly-Bernoulli numbers ahead.
