Poly-Bernoulli number

In mathematics, poly-Bernoulli numbers, denoted as $B_{n}^{(k)}$ is an integer sequence.

Definition

It was defined by 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.

Combinatorial interpretation

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 $\underbrace {1\cdots 1} _{n}\underbrace {0\cdots 0} _{k}$ (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.

See also

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

References

• Arakawa, Tsuneo; Kaneko, Masanobu (1999a), "Multiple zeta values, poly-Bernoulli numbers, and related zeta functions", Nagoya Mathematical Journal, 153: 189–209, doi:10.1017/S0027763000006954, hdl:2324/20424, MR 1684557, S2CID 53476063.
• Arakawa, Tsuneo; Kaneko, Masanobu (1999b), "On poly-Bernoulli numbers", Commentarii Mathematici Universitatis Sancti Pauli, 48 (2): 159–167, MR 1713681
• Brewbaker, Chad (2008), "A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues", Integers, 8: A02, 9, MR 2373086.
• Hamahata, Y.; Masubuchi, H. (2007), "Special multi-poly-Bernoulli numbers", Journal of Integer Sequences, 10 (4), Article 07.4.1, Bibcode:2007JIntS..10...41H, MR 2304359..