Poly-Bernoulli number explained

In mathematics, poly-Bernoulli numbers, denoted as

B ^{(k)} _{n} , were defined by M. Kaneko as{Li _{k} ( 1- e ^{-x} ) \over 1- e ^{-x} }=\sum_^B_^

where Li is the polylogarithm. The

B ^{( 1 )} _{n} 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^=\sum_^B_^(t;a,b,c)

where Li is the polylogarithm.

Kaneko also gave two combinatorial formulas:

B ^{( - k)} _{n} = \sum ^{n} _{m =0} ( -1 ) ^{m+n} m!S(n,m)(m +1 ) ^{k} , B ^{( - k)} _{n} = \sum ^{min (n,k)} _{j =0} (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 … 1 } _{n} \underbrace{ 0 … 0 } _{k} (see A329718 for definition).

The Poly-Bernoulli number

B ^{( - k)} _{k} satisfies the following asymptotic:^[1] B ^{( - k)} _{k} \sim (k!) ^{2} \sqrt{ (1)/(k\pi( 1-log2 )) }\left(\frac \right) ^, \quad \text k \rightarrow \infty.

For a positive integer n and a prime number p, the poly-Bernoulli numbers satisfy

B ^{( - p)} _{n} \equiv 2 ^{n} \pmod p,

which can be seen as an analog of Fermat's little theorem. Further, the equation

B ^{( - n)} _{x} + B ^{( - n)} _{y} = B ^{( - n)} _{z}

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

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Notes and References

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