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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 generalized hypergeometric function ： ウィキペディア英語版 generalized hypergeometric function In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. ==Notation== A hypergeometric series is formally defined as a power series :$\backslash beta\_0\; +\; \backslash beta\_1\; z\; +\; \backslash beta\_2\; z^2\; +\; \backslash dots\; =\; \backslash sum\_\; \backslash beta\_n\; z^n$ in which the ratio of successive coefficients is a rational function of ''n''. That is, :$\backslash frac\; =\; \backslash frac$ where ''A''(''n'') and ''B''(''n'') are polynomials in ''n''. For example, in the case of the series for the exponential function, :$1+\backslash frac+\backslash frac+\backslash frac+c\backslash dots,$ we have: :$\backslash beta\_n\; =\; \backslash frac,\; \backslash qquad\; \backslash frac\; =\; \backslash frac.$ So this satisfies the definition with and . It is customary to factor out the leading term, so β0 is assumed to be 1. The polynomials can be factored into linear factors of the form (''aj'' + ''n'') and (''b''''k'' + ''n'') respectively, where the ''a''''j'' and ''b''''k'' are complex numbers. For historical reasons, it is assumed that (1 + ''n'') is a factor of ''B''. If this is not already the case then both ''A'' and ''B'' can be multiplied by this factor; the factor cancels so the terms are unchanged and there is no loss of generality. The ratio between consecutive coefficients now has the form :$\backslash frac$, where ''c'' and ''d'' are the leading coefficients of ''A'' and ''B''. The series then has the form :$1\; +\; \backslash frac\backslash frac\; +\; \backslash frac\; \backslash frac\backslash left(\backslash frac\backslash right)^2+\backslash dots$, or, by scaling z by the appropriate factor and rearranging, :$1\; +\; \backslash frac\backslash frac\; +\; \backslash frac\backslash frac+\backslash dots$. This has the form of an exponential generating function. The standard notation for this series is usually denoted by: : Using the rising factorial or Pochhammer symbol: :$\backslash begin$ (a)_0 &= 1, \\ (a)_n &= a(a+1)(a+2)...(a+n-1), && n \geq 1 \end this can be written :$\backslash ,\; \backslash ,\; \backslash frac$ (Note that this use of the Pochhammer symbol is not standard, however it is the standard usage in this context.) 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「generalized hypergeometric function」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.