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 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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- Arakawa, Tsuneo; Kaneko, Masanobu (1999b), "On poly-Bernoulli numbers", Commentarii Mathematici Universitatis Sancti Pauli, 48 (2): 159–167, MR 1713681
- Brewbaker, Chad (2008), "A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues", Integers, 8: A02, 9, MR 2373086.
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- Kaneko, Masanobu (1997), "Poly-Bernoulli numbers", Journal de Théorie des Nombres de Bordeaux, 9 (1): 221–228, doi:10.5802/jtnb.197, MR 1469669.
- 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.
- Jolany, Hassan (2007), "P-adic Banach Spaces On q-Genocchi Polynomials and Generalized Poly-Bernoulli numbers" (PDF), Bachelor thesis.