Poly-Bernoulli number

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In mathematics, poly-Bernoulli numbers, denoted as , were defined by M. Kaneko as

where Li is the polylogarithm. The are the usual Bernoulli numbers.

Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined by Hassan Jolany in his bachelor thesis as follows

where Li is the polylogarithm.

Kaneko also gave two combinatorial formulas:

where is the number of ways to partition a size set into 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 by (0,1)-matrices uniquely reconstructible from their row and column sums.

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. Moreover, there is an analogue of Poly-Bernoulli numbers (like Bernoulli numbers and Euler numbers) which is known as Poly-Euler numbers

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