On congruence properties of poly-Bernoulli numbers with negative upper-indices

Abstract

For any integer $k$, M.Kaneko defined $k$-th poly-Bernoulli numbers as a kind of generalization of classical Bernoulli numbers using $k$-th polylogarithm. In case when $k$ is positive, $k$-th poly-Bernoulli numbers is a sequence of rational numbers as same as classical Bernoulli numbers. On the other hand, in case when $k$ is negative, it is a sequence of positive integers, and many combinatoric and number theoretic properties has been investigated. In the present paper, the negative case is treated, and their congruence and $p$-adic properties are discussed. Beside of them, application of the results to obtain a congruence property for the number of lonesum matrices is also mentioned.