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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 umbral calculus ： ウィキペディア英語版 umbral calculus In mathematics before the 1970s, the term ''umbral calculus'' referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to 'prove' them. These techniques were introduced by and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively.〔E. T. Bell, "The History of Blissard's Symbolic Method, with a Sketch of its Inventor's Life", ''The American Mathematical Monthly'' 45:7 (1938), pp. 414–421.〕 In the 1930s and 1940s, Eric Temple Bell attempted to set the umbral calculus on a rigorous footing. In the 1970s, Steven Roman, Gian-Carlo Rota, and others developed the umbral calculus by means of linear functionals on spaces of polynomials. Currently, ''umbral calculus'' refers to the study of Sheffer sequences, including polynomial sequences of binomial type and Appell sequences, but may encompass in its penumbra systematic correspondence techniques of the calculus of finite differences. ==The 19th-century umbral calculus== The method is a notational procedure used for deriving identities involving indexed sequences of numbers by ''pretending that the indices are exponents''. Construed literally, it is absurd, and yet it is successful: identities derived via the umbral calculus can also be properly derived by more complicated methods that can be taken literally without logical difficulty. An example involves the Bernoulli polynomials. Consider, for example, the ordinary binomial expansion (which contains a binomial coefficient): :$(y+x)^n=\backslash sum\_^ny^\; x^k$ and the remarkably similar-looking relation on the Bernoulli polynomials: :$B\_n(y+x)=\backslash sum\_^nB\_(y)\; x^k.$ Compare also the ordinary derivative :$\backslash frac\; x^n\; =\; nx^$ to a very similar-looking relation on the Bernoulli polynomials: :$\backslash frac\; B\_n(x)\; =\; nB\_(x).$ These similarities allow one to construct ''umbral'' proofs, which, on the surface, cannot be correct, but seem to work anyway. Thus, for example, by pretending that the subscript ''n'' − ''k'' is an exponent: :$B\_n(x)=\backslash sum\_^n\; b^x^k=(b+x)^n,$ and then differentiating, one gets the desired result: :$B\_n\text{'}(x)=n(b+x)^=nB\_(x).\backslash ,$ In the above, the variable ''b'' is an "umbra" (Latin for ''shadow''). See also Faulhaber'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「umbral calculus」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.