In mathematics, poly-Bernoulli numbers, denoted as , were defined by M. Kaneko as
Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined by Hassan Jolany 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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- Jolany, Hassan; Corcino, Roberto B.; Komatsu, Takao (2015), "More properties on multi-poly-Euler polynomials", Boletín de la Sociedad Matemática Mexicana 21 (2), doi:10.1007/s40590-015-0061-y.