Poly Bernoulli number - Alchetron, The Free Social Encyclopedia

In mathematics, poly-Bernoulli numbers, denoted as 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 !

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

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 !

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 , B n ( − k ) = ∑ 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 ) ≡ 2 n ( 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

References

Poly-Bernoulli number Wikipedia

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