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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 Ramanujan–Sato series ： ウィキペディア英語版 Ramanujan–Sato series In mathematics, a Ramanujan–Sato series〔Heng Huat Chan, Song Heng Chan, and Zhiguo Liu, "Domb's numbers and Ramanujan–Sato type series for 1/Pi" (2004)〕〔Gert Almkvist and Jesus Guillera, Ramanujan–Sato Like Series (2012)〕 generalizes Ramanujan’s pi formulas such as, :$\backslash frac\; =\; \backslash frac\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash frac\; =\; \backslash sum\_^\backslash infty\; s(k)\; \backslash frac$ by using other well-defined sequences of integers $s(k)$ obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients $\backslash tbinom$, and employing modular forms of higher levels. Ramanujan made the enigmatic remark that there were "corresponding theories", but it was only recently that H.H. Chan and S. Cooper found a general approach that used the underlying modular congruence subgroup $\backslash Gamma\_0(n)$,〔H.H. Chan and S. Cooper, "Rational analogues of Ramanujan's series for 1/π", Mathematical Proceedings of the Cambridge Philosophical Society / Volume 153 / Issue 02 / September 2012, pp. 361–383〕 while G. Almkvist has experimentally found numerous other examples also with a general method using differential operators.〔G. Almkvist, Some conjectured formulas for 1/Pi coming from polytopes, K3-surfaces and Moonshine, http://arxiv.org/abs/1211.6563〕 Levels ''1–4A'' were given by Ramanujan (1917),〔S. Ramanujan, "Modular equations and approximations to pi", Quart. J. Math. (Oxford) 45 (1914)〕 level ''5'' by H.H. Chan and S. Cooper (2012),〔 ''6A'' by Chan, Tanigawa, Yang, and Zudilin,〔Chan, Tanigawa, Yang, and Zudilin, "New analogues of Clausen’s identities arising from the theory of modular forms" (2011)〕 ''6B'' by Sato (2002),〔T. Sato, "Apéry numbers and Ramanujan's series for 1/π", Abstract of a talk presented at the Annual meeting of the Mathematical Society of Japan, 2002〕 ''6C'' by H. Chan, S. Chan, and Z. Liu (2004),〔 ''6D'' by H. Chan and H. Verrill (2009),〔H. Chan and H. Verrill, "The Apéry numbers, the Almkvist–Zudilin Numbers, and new series for 1/π", Advances in Mathematics, Vol 186, 2004〕 level ''7'' by S. Cooper (2012),〔S. Cooper, "Sporadic sequences, modular forms and new series for 1/π", Ramanujan Journal 2012〕 part of level ''8'' by Almkvist and Guillera (2012),〔 part of level ''10'' by Y. Yang, and the rest by H.H. Chan and S.Cooper. The notation ''j''''n''(''τ'') is derived from Zagier〔D. Zagier, "Traces of Singular Moduli", (p.15-16), http://people.mpim-bonn.mpg.de/zagier/files/tex/TracesSingModuli/fulltext.pdf〕 and ''T''''n'' refers to the relevant (McKay–Thompson series ). ==Level 1== Examples for levels 1–4 were given by Ramanujan in his 1917 paper. Given $q=e^$ as in the rest of this article. Let, :$\backslash begin$ j(\tau) &= \Big(\tfrac\Big)^3 = \tfrac + 744 + 196884q + 21493760q^2 +\dots\\ j^ *(\tau) &= 432\,\frac}} = \tfrac - 120 + 10260q - 901120q^2 + \dots \end with the j-function ''j''(''τ''), Eisenstein series ''E''4, and Dedekind eta function ''η''(''τ''). The first expansion is the McKay–Thompson series of class 1A () with a(0) = 744. Note that, as first noticed by J. McKay, the coefficient of the linear term of ''j''(''τ'') is exceedingly close to $196883$ which is the smallest degree > 1 of the irreducible representations of the Monster group. Similar phenomena will be observed in the other levels. Define, :$s\_(k)=\backslash tbinom\backslash tbinom\backslash tbinom=1,\; 120,\; 83160,\; 81681600,\backslash dots$ () :$s\_(k)=\backslash sum\_^k\backslash tbinom\backslash tbinom\backslash tbinom\backslash tbinom(-432)^\; =1,\; -312,\; 114264,\; -44196288,\backslash dots$ Then the two modular functions and sequences are related by, :$\backslash sum\_^\backslash infty\; s\_(k)\backslash ,\backslash frac^\backslash infty\; s\_(k)\backslash ,\backslash frac\; =\; 12\backslash ,\backslash boldsymbol\backslash ,\backslash sum\_^\backslash infty\; s\_(k)\backslash ,\backslash frac\}\backslash Big)=-640320^3$ :$\backslash frac\; =\; 24\backslash ,\backslash boldsymbol\backslash ,\backslash sum\_^\backslash infty\; s\_(k)\backslash ,\backslash frac)\}\},\backslash quad\; j^$ *\Big(\tfrac\Big)=-432\Big(\tfrac\Big)^=-432\,U_^ and $U\_n$ is a fundamental unit. The first belongs to a family of formulas which were rigorously proven by the Chudnovsky brothers in 1989〔.〕 and later used to calculate 10 trillion digits of π in 2011.〔.〕 The second formula, and the ones for higher levels, was established by H.H. Chan and S. Cooper in 2012.〔 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「Ramanujan–Sato series」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.