Poly-Bernoulli number

Integer sequence

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

L i k ( 1 − e − x ) 1 − e − x = ∑ n = 0 ∞ B n ( k ) x n n ! {\displaystyle {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 ) {\displaystyle B_{n}^{(1)}} are the usual Bernoulli numbers.

Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined as follows

L i k ( 1 − ( a b ) − x ) b x − a − x c x t = ∑ n = 0 ∞ B n ( k ) ( t ; a , b , c ) x n n ! {\displaystyle {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 ) = ∑ m = 0 n ( − 1 ) m + n m ! S ( n , m ) ( m + 1 ) k , {\displaystyle B_{n}^{(-k)}=\sum _{m=0}^{n}(-1)^{m+n}m!S(n,m)(m+1)^{k},}
B n ( − k ) = ∑ j = 0 min ( n , k ) ( j ! ) 2 S ( n + 1 , j + 1 ) S ( k + 1 , j + 1 ) , {\displaystyle 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 ) {\displaystyle S(n,k)} is the number of ways to partition a size n {\displaystyle n} set into k {\displaystyle 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 {\displaystyle n} by k {\displaystyle 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 1 ⋯ 1 ⏟ n 0 ⋯ 0 ⏟ k {\displaystyle \underbrace {1\cdots 1} _{n}\underbrace {0\cdots 0} _{k}} (see A329718 for definition).

The Poly-Bernoulli number B k ( − k ) {\displaystyle B_{k}^{(-k)}} satisfies the following asymptotic:[1]

B k ( − k ) ∼ ( k ! ) 2 1 k π ( 1 − log ⁡ 2 ) ( 1 log ⁡ 2 ) 2 k + 1 , as  k → ∞ . {\displaystyle 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 ) ≡ 2 n ( mod p ) , {\displaystyle 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 ) {\displaystyle 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

References

  1. ^ Khera, J.; Lundberg, E.; Melczer, S. (2021), "Asymptotic Enumeration of Lonesum Matrices", Advances in Applied Mathematics, 123 (4): 102118, arXiv:1912.08850, doi:10.1016/j.aam.2020.102118, S2CID 209414619.
  • 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.
  • Kaneko, Masanobu (1997), "Poly-Bernoulli numbers", Journal de Théorie des Nombres de Bordeaux, 9 (1): 221–228, doi:10.5802/jtnb.197, hdl:2324/21658, MR 1469669.