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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 Holonomic function ： ウィキペディア英語版 Holonomic function In mathematics, a holonomic function is a smooth function in several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonomic function is an element of a holonomic module of smooth functions. Holonomic functions can also be described as differentiably finite functions, also known as D-finite functions. When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called ''holonomic''. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial coefficients, is holonomic. ==Holonomic functions and sequences in one variable== Let $\backslash mathbb$ be a field of characteristic 0 (for example, $\backslash mathbb\; =\; \backslash mathbb$ or $\backslash mathbb\; =\; \backslash mathbb$). A function $f\; =\; f(x)$ is called ''D-finite'' (or ''holonomic'') if there exist polynomials $a\_r(x),\; a\_(x),\; \backslash ldots,\; a\_0(x)\; \backslash in\; \backslash mathbb()$ such that :$a\_r(x)\; f^(x)\; +\; a\_(x)\; f^(x)\; +\; \backslash ldots\; +\; a\_1(x)\; f\text{'}(x)\; +\; a\_0(x)\; f(x)\; =\; 0$ holds for all ''x''. This can also be written as $A\; f\; =\; 0$ where :$A\; =\; \backslash sum\_^r\; a\_k\; D\_x^k$ and $D\_x$ is the differential operator that maps $f(x)$ to $f\text{'}(x)$. $A$ is called an ''annihilating operator'' of ''f'' (the annihilating operators of $f$ form an ideal in the ring $\backslash mathbb()()$, called the ''annihilator'' of $f$). The quantity ''r'' is called the ''order'' of the annihilating operator (by extension, the sequence ''c'' is said to have order ''r'' when an annihilating operator of such order exists). A sequence $c\; =\; c\_0,\; c\_1,\; \backslash ldots$ is called ''P-recursive'' (or ''holonomic'') if there exist polynomials $a\_r(n),\; a\_(n),\; \backslash ldots,\; a\_0(n)\; \backslash in\; \backslash mathbb()$ such that :$a\_r(n)\; c\_\; +\; a\_(n)\; c\_\; +\; \backslash ldots\; +\; a\_0(n)\; c\_n\; =\; 0$ holds for all ''n''. This can also be written as $A\; c\; =\; 0$ where :$A\; =\; \backslash sum\_^r\; a\_k\; S\_n$ and $S\_n$ the shift operator that maps $c\_0,\; c\_1,\; \backslash ldots$ to $c\_1,\; c\_2,\; \backslash ldots$. $A$ is called an ''annihilating operator'' of ''c'' (the annihilating operators of $c$ form an ideal in the ring $\backslash mathbb()()$, called the ''annihilator'' of $c$). The quantity ''r'' is called the ''order'' of the annihilating operator (by extension, the sequence ''c'' is said to have order ''r'' when an annihilating operator of such order exists). Holonomic functions are precisely the generating functions of holonomic sequences: if $f(x)$ is holonomic, then the coefficients $c\_n$ in the power series expansion :$f(x)\; =\; \backslash sum\_^\; c\_n\; x^n$ form a holonomic sequence. Conversely, for a given holonomic sequence $c\_n$, the function defined by the above sum is holonomic (this is true in the sense of formal power series, even if the sum has a zero radius of convergence). 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「Holonomic function」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.