Poly-Bernoulli number

{{Short description|Integer sequence}}

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

:$\{Li\_\{k\}(1-e^\{-x\})\; \backslash over\; 1-e^\{-x\}\}=\backslash sum\_\{n=0\}^\{\backslash infty\}B\_\{n\}^\{(k)\}\{x^\{n\}\backslash over\; n!\}$

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

:$\{Li\_\{k\}(1-(ab)^\{-x\})\backslash over\; b^x-a^\{-x\}\}c^\{xt\}=\backslash sum\_\{n=0\}^\{\backslash infty\}B\_\{n\}^\{(k)\}(t;a,b,c)\{x^\{n\}\backslash over\; n!\}$

where Li is the polylogarithm.

Kaneko also gave two combinatorial formulas:

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

:$B\_\{n\}^\{(-k)\}=\backslash sum\_\{j=0\}^\{\backslash min(n,k)\}\; (j!)^\{2\}S(n+1,j+1)S(k+1,j+1),$

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 $\backslash underbrace\{1\backslash cdots1\}\_\{n\}\backslash underbrace\{0\backslash cdots0\}\_\{k\}$ (see A329718 for definition).

The Poly-Bernoulli number $B\_\{k\}^\{(-k)\}$ satisfies the following asymptotic:{{citation

| last1 = Khera | first1 = J.

| last2 = Lundberg | first2 = E.

| last3 = Melczer | first3 = S.

| issue = 4

| journal = Advances in Applied Mathematics

| title = Asymptotic Enumeration of Lonesum Matrices

| url = https://www.sciencedirect.com/science/article/abs/pii/S0196885820301214

| volume = 123

| year = 2021| page = 102118

| doi = 10.1016/j.aam.2020.102118

| arxiv = 1912.08850

| s2cid = 209414619

}}.

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

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

:$B\_n^\{(-p)\}\; \backslash equiv\; 2^n\; \backslash pmod\; p,$

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.