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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 Pochhammer k-symbol ： ウィキペディア英語版 Pochhammer k-symbol In the mathematical theory of special functions, the Pochhammer ''k''-symbol and the ''k''-gamma function, introduced by Rafael Díaz and Eddy Pariguan,〔 〕 are generalizations of the Pochhammer symbol and gamma function. They differ from the Pochhammer symbol and gamma function in that they can be related to a general arithmetic progression in the same manner as those are related to the sequence of consecutive integers. The Pochhammer ''k''-symbol (''x'')''n,k'' is defined as : $(x)\_\; =\; x(x\; +\; k)(x\; +\; 2k)\; \backslash cdots\; (x\; +\; (n-1)k)=\backslash prod\_^n\; (x+(i-1)k),\backslash ,$ and the ''k''-gamma function Γ''k'', with ''k'' > 0, is defined as : $\backslash Gamma\_k(x)\; =\; \backslash lim\_\; \backslash frac\}.$ When ''k'' = 1 the standard Pochhammer symbol and gamma function are obtained. Díaz and Pariguan use these definitions to demonstrate a number of properties of the hypergeometric function. Although Díaz and Pariguan restrict these symbols to ''k'' > 0, the Pochhammer ''k''-symbol as they define it is well-defined for all real ''k,'' and for negative ''k'' gives the falling factorial, while for ''k'' = 0 it reduces to the power ''xn''. The Díaz and Pariguan paper does not address the many analogies between the Pochhammer ''k''-symbol and the power function, such as the fact that the binomial theorem can be extended to Pochhammer ''k''-symbols. It is true, however, that many equations involving the power function ''xn'' continue to hold when ''xn'' is replaced by (''x'')''n,k''. ==References== 〔 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「Pochhammer k-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.