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 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. Also it is the number of open tours by a biased rook on a board $$\underbrace{1\cdots1}_{n}\underbrace{0\cdots0}_{k}$$ (see A329718 for definition).

The Poly-Bernoulli number $$B_{k}^{(-k)}$$ satisfies the following asymptotic:

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

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


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

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. Moreover, there is an analogue of Poly-Bernoulli numbers (like Bernoulli numbers and Euler numbers) which is known as Poly-Euler numbers.