Poly-Bernoulli number

In mathematics, poly-Bernoulli numbers, denoted as $B_^$, were defined by M. Kaneko as

:$=\sum_^B_^$

where ''Li'' is the polylogarithm. The $B_^$ are the usual Bernoulli numbers.

Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined as follows
:$c^=\sum_^B_^(t;a,b,c)$

where ''Li'' is the polylogarithm.

Kaneko also gave two combinatorial formulas:

:$B_^=\sum_^(-1)^m!S(n,m)(m+1)^,$

:$B_^=\sum_^ (j!)^S(n+1,j+1)S(k+1,j+1),$

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. Also it is the number of open tours by a biased rook on a board $\underbrace_\underbrace_$ (see A329718 for definition).

The Poly-Bernoulli number $B_^$ satisfies the following asymptotic:.

$B_^ \sim (k!)^2 \sqrt\left( \frac \right) ^, \quad \text k \rightarrow \infty.$

For a positive integer ''n'' and a prime number ''p'', the poly-Bernoulli numbers satisfy

:$B_n^ \equiv 2^n \pmod p,$

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

:$B_x^ + B_y^ = B_z^$

has no solution for integers ''x'', ''y'', ''z'', ''n'' > 2; an analog of