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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-exponential ： ウィキペディア英語版 Q-exponential In combinatorial mathematics, a ''q''-exponential is a ''q''-analog of the exponential function, namely the eigenfunction of a q-derivative. There are many q-derivatives, for example, the classical ''q''-derivative, the Askey-Wilson operator, etc. Therefore, unlike the classical exponentials, q-exponentials are not unique. For example, $e\_q(z)$ is the q-exponential corresponding to the classical ''q''-derivative while $\backslash mathcal\_q(z)$ are eigenfunctions of the Askey-Wilson operators. Interested readers may consult the reference books by M. Ismail or G. Gasper and M. Rahman provided at the end of this article. ==Definition== The ''q''-exponential $e\_q(z)$ is defined as :$e\_q(z)=$ \sum_^\infty \frac = \sum_^\infty \frac = \sum_^\infty z^n\frac where $()\_q!$ is the ''q''-factorial and :$(q;q)\_n=(1-q^n)(1-q^)\backslash cdots\; (1-q)$ is the ''q''-Pochhammer symbol. That this is the ''q''-analog of the exponential follows from the property :$\backslash left(\backslash frac\backslash right)\_q\; e\_q(z)\; =\; e\_q(z)$ where the derivative on the left is the ''q''-derivative. The above is easily verified by considering the ''q''-derivative of the monomial :$\backslash left(\backslash frac\backslash right)\_q\; z^n\; =\; z^\; \backslash frac$ =()_q z^. Here, $()\_q$ is the ''q''-bracket. 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-exponential」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.