Poly-Bernoulli number

Template:Short description In mathematics, poly-Bernoulli numbers, denoted as $B_n^{(k)}$, were defined by M. Kaneko as

$\frac{L{i}_k(1-e^{-x})}{1-e^{-x}}=\sum _{n=0}^{\mathrm{\infty }}{B}_n^{(k)}\frac{x^n}{n!}$

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

$\frac{L{i}_k(1-(ab{)}^{-x})}{b^x-a^{-x}}{c}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{B}_n^{(k)}(t;a,b,c)\frac{x^n}{n!}$

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 $\underset{n}{\underbrace{1\cdots 1}}\underset{k}{\underbrace{0\cdots 0}}$ (see A329718 for definition).

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

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

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

$B_n^{(-p)}\equiv 2^n(\mathrm{mod}p),$

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.

See also

• Bernoulli numbers
• Stirling numbers
• Gregory coefficients
• Bernoulli polynomials
• Bernoulli polynomials of the second kind
• Stirling polynomials

References

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