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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 Q-Pochhammer symbol ： ウィキペディア英語版 Q-Pochhammer symbol In mathematics, in the area of combinatorics, a ''q''-Pochhammer symbol, also called a ''q''-shifted factorial, is a ''q''-analog of the common Pochhammer symbol. It is defined as :$(a;q)\_n\; =\; \backslash prod\_^\; (1-aq^k)=(1-a)(1-aq)(1-aq^2)\backslash cdots(1-aq^)$ with :$(a;q)\_0\; =\; 1$ by definition. The ''q''-Pochhammer symbol is a major building block in the construction of ''q''-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike the ordinary Pochhammer symbol, the ''q''-Pochhammer symbol can be extended to an infinite product: :$(a;q)\_\backslash infty\; =\; \backslash prod\_^\; (1-aq^k).$ This is an analytic function of ''q'' in the interior of the unit disk, and can also be considered as a formal power series in ''q''. The special case :$\backslash phi(q)\; =\; (q;q)\_\backslash infty=\backslash prod\_^\backslash infty\; (1-q^k)$ is known as Euler's function, and is important in combinatorics, number theory, and the theory of modular forms. ==Identities== The finite product can be expressed in terms of the infinite product: :$(a;q)\_n\; =\; \backslash frac\; ,$ which extends the definition to negative integers ''n''. Thus, for nonnegative ''n'', one has :$(a;q)\_\; =\; \backslash frac=\backslash prod\_^n\; \backslash frac$ and :$(a;q)\_\; =\; \backslash frac.$ The ''q''-Pochhammer symbol is the subject of a number of ''q''-series identities, particularly the infinite series expansions :$(x;q)\_\backslash infty\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; x^n$ and :$\backslash frac=\backslash sum\_^\backslash infty\; \backslash frac$, which are both special cases of the q-binomial theorem: :$\backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; x^n.$ 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「Q-Pochhammer symbol」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.