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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 Touchard polynomials ： ウィキペディア英語版 Touchard polynomials The Touchard polynomials, studied by , also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by :$T\_n(x)=\backslash sum\_^n\; S(n,k)x^k=\backslash sum\_^n$ \left\x^k, where $S(n,k)=\backslash left\backslash$ is a Stirling number of the second kind, i.e., the number of partitions of a set of size ''n'' into ''k'' disjoint non-empty subsets. == Properties == The value at 1 of the ''n''th Touchard polynomial is the ''n''th Bell number, i.e., the number of partitions of a set of size ''n'': :$T\_n(1)=B\_n.$ If ''X'' is a random variable with a Poisson distribution with expected value λ, then its ''n''th moment is E(''X''''n'') = ''T''''n''(λ), leading to the definition: :$T\_(x)=e^\backslash sum\_^\backslash infty\; \backslash frac\; .$ Using this fact one can quickly prove that this polynomial sequence is of binomial type, i.e., it satisfies the sequence of identities: :$T\_n(\backslash lambda+\backslash mu)=\backslash sum\_^n\; T\_k(\backslash lambda)\; T\_(\backslash mu).$ The Touchard polynomials constitute the only polynomial sequence of binomial type with the coefficient of ''x'' equal 1 in every polynomial. The Touchard polynomials satisfy the Rodrigues-like formula: :$T\_n\; \backslash left(e^x\; \backslash right)\; =\; e^\; \backslash frac\backslash left(e^\backslash right)$ The Touchard polynomials satisfy the recurrence relation :$T\_(x)=x\; \backslash left(1+\backslash frac\; \backslash right)T\_(x)$ and :$T\_(x)=x\backslash sum\_^nT\_k(x).$ In the case ''x'' = 1, this reduces to the recurrence formula for the Bell numbers. Using the umbral notation ''T''''n''(''x'')=''T''''n''(''x''), these formulas become: :$T\_n(\backslash lambda+\backslash mu)=\backslash left(T(\backslash lambda)+T(\backslash mu)\; \backslash right)^n,$ :$T\_(x)=x\; \backslash left(1+T(x)\; \backslash right)^n.$ The generating function of the Touchard polynomials is :$\backslash sum\_^\backslash infty\; t^n=e^,$ which corresponds to the generating function of Stirling numbers of the second kind. Touchard polynomials have contour integral representation: :$T\_n(x)=\backslash frac\backslash oint\backslash frac\}t.$ 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「Touchard 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.