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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 Lauricella hypergeometric series ： ウィキペディア英語版 Lauricella hypergeometric series In 1893 Giuseppe Lauricella defined and studied four hypergeometric series ''F''''A'', ''F''''B'', ''F''''C'', ''F''''D'' of three variables. They are : :$$ F_A^(a,b_1,b_2,b_3,c_1,c_2,c_3;x_1,x_2,x_3) = \sum_^ \frac (b_2)_ (b_3)_} (c_3)_ \,i_1! \,i_2! \,i_3!} \,x_1^x_2^x_3^ for |''x''1| + |''x''2| + |''x''3| < 1 and :$$ F_B^(a_1,a_2,a_3,b_1,b_2,b_3,c;x_1,x_2,x_3) = \sum_^ \frac (a_3)_ (b_1)_ (b_2)_ (b_3)_} \,x_1^x_2^x_3^ for |''x''1| < 1, |''x''2| < 1, |''x''3| < 1 and :$$ F_C^(a,b,c_1,c_2,c_3;x_1,x_2,x_3) = \sum_^ \frac} (c_3)_ \,i_1! \,i_2! \,i_3!} \,x_1^x_2^x_3^ for |''x''1|½ + |''x''2|½ + |''x''3|½ < 1 and :$$ F_D^(a,b_1,b_2,b_3,c;x_1,x_2,x_3) = \sum_^ \frac (b_2)_ (b_3)_} \,x_1^x_2^x_3^ for |''x''1| < 1, |''x''2| < 1, |''x''3| < 1. Here the Pochhammer symbol (''q'')''i'' indicates the ''i''-th rising factorial of ''q'', i.e. :$(q)\_i\; =\; q\backslash ,(q+1)\; \backslash cdots\; (q+i-1)\; =\; \backslash frac~,$ where the second equality is true for all complex $q$ except $q=0,-1,-2,\backslash ldots$. These functions can be extended to other values of the variables ''x''1, ''x''2, ''x''3 by means of analytic continuation. Lauricella also indicated the existence of ten other hypergeometric functions of three variables. These were named ''F''''E'', ''F''''F'', ..., ''F''''T'' and studied by Shanti Saran in 1954 . There are therefore a total of 14 Lauricella–Saran hypergeometric functions. ==Generalization to ''n'' variables== These functions can be straightforwardly extended to ''n'' variables. One writes for example :$$ F_A^(a, b_1,\ldots,b_n, c_1,\ldots,c_n; x_1,\ldots,x_n) = \sum_^ \frac \cdots (b_n)_} \,i_1! \cdots \,i_n!} \,x_1^ \cdots x_n^ ~, where |''x''1| + ... + |''x''''n''| < 1. These generalized series too are sometimes referred to as Lauricella functions. When ''n'' = 2, the Lauricella functions correspond to the Appell hypergeometric series of two variables: :$$ F_A^ \equiv F_2 ,\quad F_B^ \equiv F_3 ,\quad F_C^ \equiv F_4 ,\quad F_D^ \equiv F_1. When ''n'' = 1, all four functions reduce to the Gauss hypergeometric function: :$$ F_A^(a,b,c;x) \equiv F_B^(a,b,c;x) \equiv F_C^(a,b,c;x) \equiv F_D^(a,b,c;x) \equiv F_1(a,b;c;x). 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「Lauricella hypergeometric 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.