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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Poly-Bernoulli number ： ウィキペディア英語版 Poly-Bernoulli number In mathematics, poly-Bernoulli numbers, denoted as $B\_^$, were defined by M. Kaneko as :$)\; \backslash over\; 1-e^\}=\backslash sum\_^B\_^$ where ''Li'' is the polylogarithm. The $B\_^$ are the usual Bernoulli numbers. Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined by Hassan Jolany as follows :$)\backslash over\; b^x-a^\}c^=\backslash sum\_^B\_^(t;a,b,c)$ where ''Li'' is the polylogarithm. Kaneko also gave two combinatorial formulas: :$B\_^=\backslash sum\_^(-1)^m!S(n,m)(m+1)^,$ :$B\_^=\backslash sum\_^\; (j!)^S(n+1,j+1)S(k+1,j+1),$ 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). 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 :$B\_n^\; \backslash equiv\; 2^n\; \backslash pmod\; p,$ which can be seen as an analog of Fermat's little theorem. 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. Moreover, there is an analogue of Poly-Bernoulli numbers (like Bernoulli numbers and Euler numbers) which is known as Poly-Euler numbers ==References== *. * *. *. *. *. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Poly-Bernoulli number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.