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.
- Hassan Jolany, Explicit formula for generalization of Poly-Bernoulli numbers and polynomials with a,b,c parameters, ,2012
- M. Kaneko, Poly-Bernoulli numbers, Journal de Theorie des Nombres de Bordeaux, 9:221-228, 1997
- 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