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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 symbol ： ウィキペディア英語版 Pochhammer symbol In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation , where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which interpretation is being used in any particular article. Pochhammer himself actually used with yet another meaning, namely to denote the binomial coefficient $\backslash tbinom\; xn$.〔. The remark about the Pochhammer symbol is on page 414.〕 In this article the Pochhammer symbol is used to represent the falling factorial (sometimes called the ''"descending factorial"'',〔 ''"falling sequential product"'', ''"lower factorial"''): :$(x)\_=x(x-1)(x-2)\backslash cdots(x-n+1)$ In this article the symbol is used for the rising factorial (sometimes called the ''"Pochhammer function"'', ''"Pochhammer polynomial"'', ''"ascending factorial"'',〔 (A reprint of the 1950 edition by Chelsea Publishing Co.)〕 ''"rising sequential product"'' or ''"upper factorial"''): :$x^=x(x+1)(x+2)\backslash cdots(x+n-1).$ These conventions are used in combinatorics. However in the theory of special functions (in particular the hypergeometric function) the Pochhammer symbol is used to represent the rising factorial.〔so is the case in Abramowitz and Stegun's "Handbook of Mathematical Functions", P. 256〕 A useful list of formulas for manipulating the rising factorial in this last notation is given in . Knuth uses the term factorial powers to comprise rising and falling factorials.〔Knuth, The Art of Computer Programming, Vol. 1, 3rd ed., p. 50.〕 When is a non-negative integer, then gives the number of of an -element set, or equivalently the number of injective functions from a set of size to a set of size . However, for these meanings other notations like and ''P''(''x,n'') are commonly used. The Pochhammer symbol serves mostly for more algebraic uses, for instance when is an indeterminate, in which case designates a particular polynomial of degree in . ==Properties== The rising and falling factorials can be used to express a binomial coefficient: :$\backslash frac\; =\; \backslash quad\backslash mbox\backslash quad\; \backslash frac\; =\; .$ Thus many identities on binomial coefficients carry over to the falling and rising factorials. A rising factorial can be expressed as a falling factorial that starts from the other end, :$x^\; =\; \_n\; ,$ or as a falling factorial with opposite argument, :$x^\; =\; ^n\; \_\; .$ The rising and falling factorials are well defined in any unital ring, and therefore ''x'' can be taken to be, for example, a complex number, including negative integers, or a polynomial with complex coefficients, or any complex-valued function. The rising factorial can be extended to real values of using the Gamma function provided and are complex numbers that are not negative integers: :$x^=\backslash frac,$ and so can the falling factorial: :$(x)\_n=\backslash frac.$ If denotes differentiation with respect to , one has :$D^n(x^a)\; =\; (a)\_n\backslash ,\backslash ,\; x^.$ The Pochhammer symbol is also integral to the definition of the hypergeometric function: The hypergeometric function is defined for |''z''| < 1 by the power series :$\backslash ,\_2F\_1(a,b;c;z)\; =\; \backslash sum\_^\backslash infty\; \backslash over\; c^\}\; \backslash ,$ provided that ''c'' does not equal 0, −1, −2, ... . Note, however, that the hypergeometric function literature uses the notation $\_$ for rising factorials. 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 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.