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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 Jacobi triple product ： ウィキペディア英語版 Jacobi triple product In mathematics, the Jacobi triple product is the mathematical identity: :$\backslash prod\_^\backslash infty$ \left( 1 - x^\right) \left( 1 + x^ y^2\right) \left( 1 +\frac\right) = \sum_^\infty x^ y^, for complex numbers ''x'' and ''y'', with |''x''| < 1 and ''y'' ≠ 0. It was introduced by in his work ''Fundamenta Nova Theoriae Functionum Ellipticarum''. The Jacobi triple product identity is the Macdonald identity for the affine root system of type ''A''1, and is the Weyl denominator formula for the corresponding affine Kac–Moody algebra. == Properties == The basis of Jacobi's proof relies on Euler's pentagonal number theorem, which is itself a specific case of the Jacobi Triple Product Identity. Let $x=q\backslash sqrt\; q$ and $y^2=-\backslash sqrt$. Then we have :$\backslash phi(q)\; =\; \backslash prod\_^\backslash infty\; \backslash left(1-q^m\; \backslash right)\; =$ \sum_^\infty (-1)^n q^}.\, The Jacobi Triple Product also allows the Jacobi theta function to be written as an infinite product as follows: Let $x=e^$ and $y=e^.$ Then the Jacobi theta function :$$ \vartheta(z; \tau) = \sum_^\infty e^} n z} can be written in the form :$\backslash sum\_^\backslash infty\; y^x^.$ Using the Jacobi Triple Product Identity we can then write the theta function as the product :$\backslash vartheta(z;\; \backslash tau)\; =\; \backslash prod\_^\backslash infty$ \left( 1 - e^\right) \left(1 + e^} z}\right ) \left(1 + e^} z}\right ). There are many different notations used to express the Jacobi triple product. It takes on a concise form when expressed in terms of q-Pochhammer symbols: :$\backslash sum\_^\backslash infty\; q^\}z^n\; =$ (q;q)_\infty \; \left(-\frac;q\right)_\infty \; (-zq;q)_\infty, where $(a;q)\_\backslash infty$ is the infinite ''q''-Pochhammer symbol. It enjoys a particularly elegant form when expressed in terms of the Ramanujan theta function. For it can be written as :$\backslash sum\_^\backslash infty\; a^\}\; \backslash ;\; b^\}\; =\; (-a;\; ab)\_\backslash infty\; \backslash ;(-b;\; ab)\_\backslash infty\; \backslash ;(ab;ab)\_\backslash infty.$ 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「Jacobi triple product」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.