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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 binomial type ： ウィキペディア英語版 binomial type In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities :$p\_n(x+y)=\backslash sum\_^n\backslash ,\; p\_k(x)\backslash ,\; p\_(y).$ Many such sequences exist. The set of all such sequences forms a Lie group under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type). Polynomial sequences put on firm footing the vague 19th century notions of umbral calculus. ==Examples== * In consequence of this definition the binomial theorem can be stated by saying that the sequence is of binomial type. * The sequence of "lower factorials" is defined by ::$(x)\_n=x(x-1)(x-2)\backslash cdot\backslash cdots\backslash cdot(x-n+1).$ :(In the theory of special functions, this same notation denotes upper factorials, but this present usage is universal among combinatorialists.) The product is understood to be 1 if ''n'' = 0, since it is in that case an empty product. This polynomial sequence is of binomial type. * Similarly the "upper factorials" ::$x^=x(x+1)(x+2)\backslash cdot\backslash cdots\backslash cdot(x+n-1)$ :are a polynomial sequence of binomial type. * The Abel polynomials ::$p\_n(x)=x(x-an)^\; \backslash ,$ :are a polynomial sequence of binomial type. * The Touchard polynomials ::$p\_n(x)=\backslash sum\_^n\; S(n,k)x^k$ :where ''S''(''n'', ''k'') is the number of partitions of a set of size ''n'' into ''k'' disjoint non-empty subsets, is a polynomial sequence of binomial type. Eric Temple Bell called these the "exponential polynomials" and that term is also sometimes seen in the literature. The coefficients ''S''(''n'', ''k'' ) are "Stirling numbers of the second kind". This sequence has a curious connection with the Poisson distribution: If ''X'' is a random variable with a Poisson distribution with expected value λ then E(''X''''n'') = ''p''''n''(λ). In particular, when λ = 1, we see that the ''n''th moment of the Poisson distribution with expected value 1 is the number of partitions of a set of size ''n'', called the ''n''th Bell number. This fact about the ''n''th moment of that particular Poisson distribution is "Dobinski's formula". 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「binomial type」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.