In mathematics, poly-Bernoulli numbers, denoted as
, were defined by M. Kaneko as where Li is the polylogarithm.
are the usual Bernoulli numbers.
Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined as follows where Li is the polylogarithm.
Kaneko also gave two combinatorial formulas: where
is the number of ways to partition a size
non-empty subsets (the Stirling number of the second kind).
A combinatorial interpretation is that the poly-Bernoulli numbers of negative index enumerate the set of
(0,1)-matrices uniquely reconstructible from their row and column sums.
Also it is the number of open tours by a biased rook on a board
For a positive integer n and a prime number p, the poly-Bernoulli numbers satisfy which can be seen as an analog of Fermat's little theorem.
Further, the equation has no solution for integers x, y, z, n > 2; an analog of Fermat's Last Theorem.