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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 transform ： ウィキペディア英語版 binomial transform In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function. ==Definition== The binomial transform, ''T'', of a sequence, , is the sequence defined by :$s\_n\; =\; \backslash sum\_^n\; (-1)^k\; a\_k.$ Formally, one may write (''Ta'')''n'' = ''s''''n'' for the transformation, where ''T'' is an infinite-dimensional operator with matrix elements ''T''''nk'': :$s\_n\; =\; (Ta)\_n\; =\; \backslash sum\_^\backslash infty\; T\_\; a\_k.$ The transform is an involution, that is, :$TT\; =\; 1\; \backslash ,$ or, using index notation, :$\backslash sum\_^\backslash infty\; T\_T\_\; =\; \backslash delta\_$ where $\backslash delta\_$ is the Kronecker delta. The original series can be regained by :$a\_n=\backslash sum\_^n\; (-1)^k\; s\_k.$ The binomial transform of a sequence is just the ''n''th forward differences of the sequence, with odd differences carrying a negative sign, namely: :$s\_0\; =\; a\_0$ :$s\_1\; =\; -\; (\backslash triangle\; a)\_0\; =\; -a\_1+a\_0$ :$s\_2\; =\; (\backslash triangle^2\; a)\_0\; =\; -(-a\_2+a\_1)+(-a\_1+a\_0)\; =\; a\_2-2a\_1+a\_0$ :$\backslash vdots\backslash ,$ :$s\_n\; =\; (-1)^n\; (\backslash triangle^n\; a)\_0$ where Δ is the forward difference operator. Some authors define the binomial transform with an extra sign, so that it is not self-inverse: :$t\_n=\backslash sum\_^n\; (-1)^\; a\_k$ whose inverse is :$a\_n=\backslash sum\_^n\; t\_k.$ 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 transform」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.