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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 Gegenbauer polynomials ： ウィキペディア英語版 Gegenbauer polynomials In mathematics, Gegenbauer polynomials or ultraspherical polynomials ''C''(''x'') are orthogonal polynomials on the interval () with respect to the weight function (1 − ''x''2)''α''–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer. ==Characterizations== Mplwp gegenbauer Cn05a1.svg|Gegenbauer polynomials with ''α''=1 Mplwp gegenbauer Cn05a2.svg|Gegenbauer polynomials with ''α''=2 Mplwp gegenbauer Cn05a3.svg|Gegenbauer polynomials with ''α''=3 Gegenbauer polynomials.gif|Gegenbauer polynomials A variety of characterizations of the Gegenbauer polynomials are available. * The polynomials can be defined in terms of their generating function : ::$\backslash frac=\backslash sum\_^\backslash infty\; C\_n^(x)\; t^n.$ * The polynomials satisfy the recurrence relation : ::$$ \begin C_0^\alpha(x) & = 1 \\ C_1^\alpha(x) & = 2 \alpha x \\ C_n^\alpha(x) & = \frac(- (n+2\alpha-2)C_^\alpha(x) ). \end * Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation : ::$(1-x^)y\text{'}\text{'}-(2\backslash alpha+1)xy\text{'}+n(n+2\backslash alpha)y=0.\backslash ,$ :When ''α'' = 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the Legendre polynomials. * They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite: ::$C\_n^(z)=\backslash frac$ \,_2F_1\left(-n,2\alpha+n;\alpha+\frac;\frac\right). :(Abramowitz & Stegun (p. 561 )). Here (2α)''n'' is the rising factorial. Explicitly, ::$$ C_n^(z)=\sum_^ (-1)^k\frac(2z)^. * They are special cases of the Jacobi polynomials : ::$C\_n^(x)\; =\; \backslash frac)\_\}P\_n^(x).$ :in which $(\backslash theta)\_n$ represents the rising factorial of $\backslash theta$. :One therefore also has the Rodrigues formula ::$C\_n^(x)\; =\; \backslash frac\backslash frac(1-x^2)^\backslash frac\backslash left().$ 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「Gegenbauer polynomials」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.