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 by Hassan Jolany in his bachelor thesis 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.

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

References

Arakawa, Tsuneo; Kaneko, Masanobu (1999a), "Multiple zeta values, poly-Bernoulli numbers, and related zeta functions", Nagoya Mathematical Journal, 153: 189–209, MR 1684557 .
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, MR 2304359 .
Kaneko, Masanobu (1997), "Poly-Bernoulli numbers", Journal de Théorie des Nombres de Bordeaux, 9 (1): 221–228, doi:10.5802/jtnb.197, MR 1469669 .
Jolany, Hassan; Corcino, Roberto B.; Komatsu, Takao (2015), "More properties on multi-poly-Euler polynomials", Boletín de la Sociedad Matemática Mexicana, 21 (2), doi:10.1007/s40590-015-0061-y .
Jolany, Hassan (2007), "P-adic Banach Spaces On q-Genocchi Polynomials and Generalized Poly-Bernoulli numbers" (PDF), Bachelor thesis .

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