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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 Multiple zeta function ： ウィキペディア英語版 Multiple zeta function In mathematics, the multiple zeta functions are generalisations of the Riemann zeta function, defined by :$$ \zeta(s_1, \ldots, s_k) = \sum_ \ \frac} = \sum_ \ \prod_^k \frac^2,3) ==Two parameters case== In the particular case of only two parameters we have (with s>1 and n,m integer): :$\backslash zeta(s,t)\; =\; \backslash sum\_\; \backslash \; \backslash frac\}\; =\; \backslash sum\_^\; \backslash frac^\; \backslash frac\; =\; \backslash sum\_^\; \backslash frac^\; \backslash frac$ :$\backslash zeta(s,t)=\backslash sum\_^\backslash infty\; \backslash frac$ where $H\_$ are the generalized harmonic numbers. Multiple zeta functions are known to satisfy what is known as MZV duality, the simplest case of which is the famous identity of Euler: :$$ \sum_^\infty \frac = \zeta(2,1) = \zeta(3) = \sum_^\infty \frac, \! where ''H''''n'' are the harmonic numbers. Special values of double zeta functions, with ''s'' > 0 and even, ''t'' > 1 and odd, but s+t=2N+1 (taking if necessary ''ζ''(0) = 0):〔 :$\backslash zeta(s,t)=\backslash zeta(s)\backslash zeta(t)+\backslash tfrac\backslash Big()\backslash zeta(s+t)-\backslash sum\_^\backslash Big()\backslash zeta(2r+1)\backslash zeta(s+t-1-2r)$ \zeta(4) || |- | 3 || 2 || 0.228810397603353759768746148942 || $3\backslash zeta(2)\backslash zeta(3)-\backslash tfrac\backslash zeta(5)$ |- | 4 || 2 || 0.088483382454368714294327839086 || $\backslash left\; (\backslash zeta(3)\backslash right\; )^2-\backslash tfrac\backslash zeta(6)$ |- | 5 || 2 || 0.038575124342753255505925464373 || $5\backslash zeta(2)\backslash zeta(5)+2\backslash zeta(3)\backslash zeta(4)-11\backslash zeta(7)$ |- | 6 || 2 || 0.017819740416835988 || |- | 2 || 3 || 0.711566197550572432096973806086 || $\backslash tfrac\backslash zeta(5)-2\backslash zeta(2)\backslash zeta(3)$ |- | 3 || 3 || 0.213798868224592547099583574508 || $\backslash tfrac\backslash left\; (\backslash left\; (\backslash zeta(3)\backslash right\; )^2\; -\backslash zeta(6)\backslash right\; )$ |- | 4 || 3 || 0.085159822534833651406806018872 || $17\backslash zeta(7)-10\backslash zeta(2)\backslash zeta(5)$ |- | 5 || 3 || 0.037707672984847544011304782294 || $5\backslash zeta(3)\backslash zeta(5)-\backslash tfrac\backslash zeta(8)-\backslash tfrac\backslash zeta(6,2)$ |- | 2 || 4 || 0.674523914033968140491560608257 || $\backslash tfrac\backslash zeta(6)-\backslash left\; (\backslash zeta(3)\backslash right\; )^2$ |- | 3 || 4 || 0.207505014615732095907807605495 || $10\backslash zeta(2)\backslash zeta(5)+\backslash zeta(3)\backslash zeta(4)-18\backslash zeta(7)$ |- | 4 || 4 || 0.083673113016495361614890436542 || $\backslash tfrac\backslash left\; (\backslash left\; (\backslash zeta(4)\backslash right\; )^2\; -\backslash zeta(8)\backslash right\; )$ |} Note that if $s+t=2p+2$ we have $p/3$ irreducibles, i.e. these MZVs cannot be written as function of $\backslash zeta(a)$ only.〔 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「Multiple zeta function」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.