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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 Rogers–Ramanujan continued fraction ： ウィキペディア英語版 Rogers–Ramanujan continued fraction The Rogers–Ramanujan continued fraction is a continued fraction discovered by and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument. ==Definition== Given the functions ''G''(''q'') and ''H''(''q'') appearing in the Rogers–Ramanujan identities, :$\backslash beginG(q)$ &= \sum_^\infty \frac =\sum_^\infty \frac = \frac \\ &= \prod_^\infty \frac)}\\ &=\sqrt()\,_2F_1\left(-\tfrac,\tfrac;\tfrac;\tfrac\right)\\ &=\sqrt()\,_2F_1\left(-\tfrac,\tfrac;\tfrac;-\tfrac\right)\\ &= 1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots \end and, :$\backslash beginH(q)$ &= \sum_^\infty \frac =\sum_^\infty \frac = \frac \\ &= \prod_^\infty \frac)}\\ &=\frac}}\,_2F_1\left(\tfrac,\tfrac;\tfrac;\tfrac\right)\\ &=\frac}}\,_2F_1\left(\tfrac,\tfrac;\tfrac;-\tfrac\right)\\ &= 1+q^2 +q^3 +q^4+q^5 +2q^6+2q^7+\cdots \end and , respectively, where $(a;q)\_\backslash infty$ denotes the infinite q-Pochhammer symbol, ''j'' is the j-function, and 2F1 is the hypergeometric function, then the Rogers–Ramanujan continued fraction is, :$\backslash beginR(q)$ &= \frac}H(q)}}G(q)} = q^}\prod_^\infty \frac)})}\\ &= \cfrac}}} \end 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「Rogers–Ramanujan continued fraction」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.