Poly-Bernoulli number

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In mathematics, poly-Bernoulli numbers, denoted as B_{n}^{(k)}, were defined by M. Kaneko as

{Li_{k}(1-e^{-z}) \over 1-e^{-x}}=\sum_{n=0}^{\infty}B_{n}^{(k)}{x^{n}\over n!}

where Li is the polylogarithm. The B_{n}^{(1)} are the usual Bernoulli numbers.

Kaneko also gave two combinatorial formulas:

B_{n}^{(-k)}=\sum_{m=0}^{n}(-1)^{m+n}m!S(n,m)(m+1)^{k},
B_{n}^{(-k)}=\sum_{j=0}^{\min(n,k)} (j!)^{2}S(n+1,j+1)S(k+1,j+1),

where S(n,k) is the number of ways to partition a size n set into k 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 n by k (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

B_n^{(-p)} \equiv 2^n \pmod p,

which can be seen as an analog of Fermat's little theorem. Further, the equation

B_x^{(-n)} + B_y^{(-n)} = B_z^{(-n)}

has no solution for integers x, y, z, n > 2; an analog of Fermat's last theorem.

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