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
Poly-Bernoulli numbers have the same duality which known as Poly-Euler numbers
- Jolany, Hassan (2012). "Explicit formula for generalization of Poly-Bernoulli numbers and polynomials with a,b,c parameters". arXiv:1109.1387 [math.NT].
- Kaneko, Masanobou (1997). "Poly-Bernoulli numbers". J. Théor. Nombres Bordx. 9: 221–228. doi:10.5802/jtnb.197. Zbl 0887.11011.
- Chad Brewbaker, Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index, Master's thesis, Iowa State University, 2005
- Chad Brewbaker, A Combinatorial Interpretation of the Poly-Bernoulli Numbers and Two Fermat Analogues, INTEGERS, VOL 8, A3, 2008
- Hassan Jolany, Mohsen Aliabadi, Roberto B. Corcino, and M.R.Darafsheh, A Note On Multi Poly-Euler Numbers And Bernoulli Polynomials, ,2012