Poly-Bernoulli number

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.

References

* M. Kaneko, "Poly-Bernoulli numbers", Journal de Theorie des Nombres de Bordeaux, 9:221-228, 1997
* C. R. Brewbaker, " [http://www.public.iastate.edu/~crb002/thesis.pdf Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index] ", Master's thesis, Iowa State University, 2005