In mathematics, poly-Bernoulli numbers, denoted as B n ( k ) {\displaystyle B_{n}^{(k)}} is an integer sequence.
Definition It was defined by 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.
Combinatorial interpretation 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:
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