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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 Appell hypergeometric series ： ウィキペディア英語版 Appell series In mathematics, Appell series are a set of four hypergeometric series ''F''1, ''F''2, ''F''3, ''F''4 of two variables that were introduced by and that generalize Gauss's hypergeometric series 2''F''1 of one variable. Appell established the set of partial differential equations of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable. ==Definitions== The Appell series ''F''1 is defined for |''x''| < 1, |''y''| < 1 by the double series: :$$ F_1(a,b_1,b_2,c;x,y) = \sum_^\infty \frac \,x^m y^n ~, where the Pochhammer symbol (''q'')''n'' represents the rising factorial: :$(q)\_n\; =\; q\backslash ,(q+1)\; \backslash cdots\; (q+n-1)\; =\; \backslash frac~,$ where the second equality is true for all complex $q$ except $q=0,-1,-2,\backslash ldots$. For other values of ''x'' and ''y'' the function ''F''1 can be defined by analytic continuation. Similarly, the function ''F''2 is defined for |''x''| + |''y''| < 1 by the series: :$$ F_2(a,b_1,b_2,c_1,c_2;x,y) = \sum_^\infty \frac \,x^m y^n ~, the function ''F''3 for |''x''| < 1, |''y''| < 1 by the series: :$$ F_3(a_1,a_2,b_1,b_2,c;x,y) = \sum_^\infty \frac \,x^m y^n ~, and the function ''F''4 for |''x''|½ + |''y''|½ < 1 by the series: :$$ F_4(a,b,c_1,c_2;x,y) = \sum_^\infty \frac} \,x^m y^n ~. 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「Appell 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.