Poly-Bernoulli number

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

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

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

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

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

where Li is the polylogarithm.

Kaneko also gave two combinatorial formulas:

${\displaystyle B_{n}^{(-k)}=\sum _{m=0}^{n}(-1)^{m+n}m!S(n,m)(m+1)^{k},}$

${\displaystyle B_{n}^{(-k)}=\sum _{j=0}^{\min(n,k)}(j!)^{2}S(n+1,j+1)S(k+1,j+1),}$

where ${\displaystyle S(n,k)}$ is the number of ways to partition a size ${\displaystyle n}$ set into ${\displaystyle k}$ 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 ${\displaystyle n}$ by ${\displaystyle k}$ (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 ${\displaystyle \underbrace {1\cdots 1} _{n}\underbrace {0\cdots 0} _{k}}$ (see A329718 for definition).

The Poly-Bernoulli number ${\displaystyle B_{k}^{(-k)}}$ satisfies the following asymptotic:[1]

${\displaystyle B_{k}^{(-k)}\sim (k!)^{2}{\sqrt {\frac {1}{k\pi (1-\log 2)}}}\left({\frac {1}{\log 2}}\right)^{2k+1},\quad {\text{as }}k\rightarrow \infty .}$

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

${\displaystyle B_{n}^{(-p)}\equiv 2^{n}{\pmod {p}},}$

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

${\displaystyle B_{x}^{(-n)}+B_{y}^{(-n)}=B_{z}^{(-n)}}$

has no solution for integers x, y, z, n > 2; an analog of Fermat's Last Theorem. Moreover, there is an analogue of Poly-Bernoulli numbers (like Bernoulli numbers and Euler numbers) which is known as Poly-Euler numbers.