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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 Table of Newtonian series ： ウィキペディア英語版 Table of Newtonian series In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence $a\_n$ written in the form :$f(s)\; =\; \backslash sum\_^\backslash infty\; (-1)^n\; a\_n\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; a\_n$ where :$$ is the binomial coefficient and $(s)\_n$ is the rising factorial. Newtonian series often appear in relations of the form seen in umbral calculus. ==List== The generalized binomial theorem gives :$(1+z)^\; =\; \backslash sum\_^z^n\; =$ 1+z+z^2+\cdots. A proof for this identity can be obtained by showing that it satisfies the differential equation : $(1+z)\; \backslash frac\; =\; s\; (1+z)^s.$ The digamma function: :$\backslash psi(s+1)=-\backslash gamma-\backslash sum\_^\backslash infty\; \backslash frac\; .$ The Stirling numbers of the second kind are given by the finite sum :$\backslash left\backslash \backslash right\backslash \}$ =\frac\sum_^(-1)^ j^n. This formula is a special case of the ''k''th forward difference of the monomial ''x''''n'' evaluated at ''x'' = 0: :$\backslash Delta^k\; x^n\; =\; \backslash sum\_^(-1)^\; (x+j)^n.$ A related identity forms the basis of the Nörlund–Rice integral: :$\backslash sum\_^n\; \backslash frac\; =$ \frac = \frac= B(n+1,s-n) where $\backslash Gamma(x)$ is the Gamma function and $B(x,y)$ is the Beta function. The trigonometric functions have umbral identities: :$\backslash sum\_^\backslash infty\; (-1)^n\; =\; 2^\; \backslash cos\; \backslash frac$ and :$\backslash sum\_^\backslash infty\; (-1)^n\; =\; 2^\; \backslash sin\; \backslash frac$ The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial $(s)\_n$. The first few terms of the sin series are :$s\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots\backslash ,$ which can be recognized as resembling the Taylor series for sin ''x'', with (''s'')''n'' standing in the place of ''x''''n''. In analytic number theory it is of interest to sum :$\backslash !\backslash sum\_B\_k\; z^k,$ where ''B'' are the Bernoulli numbers. Employing the generating function its Borel sum can be evaluated as :$\backslash sum\_B\_k\; z^k=\; \backslash int\_0^\backslash infty\; e^\; \backslash fracd\; t=\; \backslash sum\_\backslash frac\; z.$ The general relation gives the Newton series :$\backslash sum\_\backslash frac\backslash frac=\; z^\backslash zeta(s,x+z),$ where $\backslash zeta$ is the Hurwitz zeta function and $B\_k(x)$ the Bernoulli polynomial. The series does not converge, the identity holds formally. Another identity is $\backslash frac\; 1=\; \backslash sum\_^\backslash infty\; \backslash sum\_^k\; \backslash frac,$ which converges for $x>a$. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent) :$f(x)=\backslash sum\_\; \backslash sum\_^k\; (-1)^f(a+j\; h).$ 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「Table of Newtonian 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.