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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 Basic hypergeometric series ： ウィキペディア英語版 Basic hypergeometric series In mathematics, Heine's basic hypergeometric series, or hypergeometric q-series, are q-analog generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x''''n'' is called hypergeometric if the ratio of successive terms ''x''''n''+1/''x''''n'' is a rational function of ''n''. If the ratio of successive terms is a rational function of ''q''''n'', then the series is called a basic hypergeometric series. The number ''q'' is called the base. The basic hypergeometric series 2φ1(''q''α,''q''β;''q''γ;''q'',''x'') was first considered by . It becomes the hypergeometric series ''F''(α,β;γ;''x'') in the limit when the base ''q'' is 1. ==Definition== There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic geometric series ψ. The unilateral basic hypergeometric series is defined as :$\backslash ;\_\backslash phi\_k\; \backslash left($ a_1 & a_2 & \ldots & a_ \\ b_1 & b_2 & \ldots & b_k \end ; q,z \right ) = \sum_^\infty \frac \left((-1)^nq^\right)^z^n where :$(a\_1,a\_2,\backslash ldots,a\_m;q)\_n\; =\; (a\_1;q)\_n\; (a\_2;q)\_n\; \backslash ldots\; (a\_m;q)\_n$ and where :$(a;q)\_n\; =\; \backslash prod\_^\; (1-aq^k)=(1-a)(1-aq)(1-aq^2)\backslash cdots(1-aq^).$ is the q-shifted factorial. The most important special case is when ''j'' = ''k''+1, when it becomes :$\backslash ;\_\backslash phi\_k\; \backslash left($ a_1 & a_2 & \ldots & a_&a_ \\ b_1 & b_2 & \ldots & b_ \end ; q,z \right ) = \sum_^\infty \frac z^n. This series is called balanced if ''a''1''...''a''''k''+1'' = ''b''1...''b''''k''''q''. This series is called well poised if ''a''1''q'' = ''a''2''b''1 = ... = ''a''k+1''b''''k'', and very well poised if in addition ''a''2 = −''a''3 = ''qa''11/2. The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series, is defined as :$\backslash ;\_j\backslash psi\_k\; \backslash left($ a_1 & a_2 & \ldots & a_j \\ b_1 & b_2 & \ldots & b_k \end ; q,z \right ) = \sum_^\infty \frac \left((-1)^nq^\right)^z^n. The most important special case is when ''j'' = ''k'', when it becomes :$\backslash ;\_k\backslash psi\_k\; \backslash left($ a_1 & a_2 & \ldots & a_k \\ b_1 & b_2 & \ldots & b_k \end ; q,z \right ) = \sum_^\infty \frac z^n. The unilateral series can be obtained as a special case of the bilateral one by setting one of the ''b'' variables equal to ''q'', at least when none of the ''a'' variables is a power of ''q''., as all the terms with ''n''<0 then vanish. 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「Basic 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.