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, poly-Bernoulli numbers, denoted as

B_^

, were defined by M. Kaneko as :

=\sum_^B_^

where ''Li'' is the

polylogarithm In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...

. The

B_^

are the usual

Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...

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

c^=\sum_^B_^(t;a,b,c)

where ''Li'' is the

. Kaneko also gave two combinatorial formulas: :

B_^=\sum_^(-1)^m!S(n,m)(m+1)^,

B_^=\sum_^ (j!)^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 In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \lef ...

). A combinatorial interpretation is that the poly-Bernoulli numbers of negative index enumerate the set of

n

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_\underbrace_

(see A329718 for definition). The Poly-Bernoulli number

B_^

satisfies the following asymptotic:.

B_^ \sim (k!)^2 \sqrt\left( \frac \right) ^, \quad \text k \rightarrow \infty.

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

B_n^ \equiv 2^n \pmod p,

which can be seen as an analog of

Fermat's little theorem Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = ...

. Further, the equation :

B_x^ + B_y^ = B_z^

has no solution for integers ''x'', ''y'', ''z'', ''n'' > 2; an analog of

Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...

. Moreover, there is an analogue of Poly-Bernoulli numbers (like Bernoulli numbers and Euler numbers) which is known as Poly-Euler numbers.

References

*. * *. *. *{{citation , last = Kaneko , first = Masanobu , doi = 10.5802/jtnb.197 , issue = 1 , journal = Journal de Théorie des Nombres de Bordeaux , mr = 1469669 , pages = 221–228 , title = Poly-Bernoulli numbers , url = http://jtnb.cedram.org/item?id=JTNB_1997__9_1_221_0 , volume = 9 , year = 1997, hdl = 2324/21658 , doi-access = free . Integer sequences Enumerative combinatorics

See also

References