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| In [[mathematics]], '''poly-Bernoulli numbers''', denoted as <math>B_{n}^{(k)}</math>, were defined by M. Kaneko as
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| :<math>{Li_{k}(1-e^{-x}) \over 1-e^{-x}}=\sum_{n=0}^{\infty}B_{n}^{(k)}{x^{n}\over n!}</math>
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| where ''Li'' is the [[polylogarithm]]. The <math>B_{n}^{(1)}</math> are the usual [[Bernoulli number]]s.
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| Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined by H. Jolany as follows
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| :<math>{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!}</math>
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| where ''Li'' is the [[polylogarithm]].
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| Kaneko also gave two combinatorial formulas:
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| :<math>B_{n}^{(-k)}=\sum_{m=0}^{n}(-1)^{m+n}m!S(n,m)(m+1)^{k},</math>
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| :<math>B_{n}^{(-k)}=\sum_{j=0}^{\min(n,k)} (j!)^{2}S(n+1,j+1)S(k+1,j+1),</math>
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| where <math>S(n,k)</math> is the number of ways to partition a size <math>n</math> set into <math>k</math> non-empty subsets (the [[Stirling number of the second kind]]).
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| A combinatorial interpretation is that the poly-Bernoulli numbers of negative index enumerate the set of <math>n</math> by <math>k</math> [[binary matrix|(0,1)-matrices]] uniquely reconstructible from their row and column sums.
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| For a positive integer ''n'' and a prime number ''p'', the poly-Bernoulli numbers satisfy
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| :<math>B_n^{(-p)} \equiv 2^n \pmod p,</math>
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| which can be seen as an analog of [[Fermat's little theorem]]. Further, the equation
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| :<math>B_x^{(-n)} + B_y^{(-n)} = B_z^{(-n)}</math>
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| has no solution for integers ''x'', ''y'', ''z'', ''n'' > 2; an analog of [[Fermat's last theorem]].
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| ==References==
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| * Hassan Jolany, Explicit formula for generalization of Poly-Bernoulli numbers and polynomials with a,b,c parameters, ''[http://arxiv.org/abs/1109.1387 ]'',2012
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| * M. Kaneko, ''Poly-Bernoulli numbers'', Journal de Theorie des Nombres de Bordeaux, 9:221-228, 1997
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| * Chad Brewbaker, ''[http://www.public.iastate.edu/~crb002/thesis.pdf Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index]'', Master's thesis, Iowa State University, 2005
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| * Chad Brewbaker, A Combinatorial Interpretation of the Poly-Bernoulli Numbers and Two Fermat Analogues, INTEGERS, [http://www.integers-ejcnt.org/vol8.html VOL 8], A3, 2008
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| [[Category:Integer sequences]]
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| [[Category:Enumerative combinatorics]]
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