Poly-Bernoulli number

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

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

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

Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined by Hassan Jolany as follows

${Li_{k}(1-(ab)^{-x})\over b^x-a^{-x}}c^{xt}=\sum_{n=0}^{\infty}B_{n}^{(k)}(t;a,b,c){x^{n}\over n!}$

where Li is the polylogarithm. 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.

References

• Hassan Jolany, Explicit formula for generalization of Poly-Bernoulli numbers and polynomials with a,b,c parameters, [1],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