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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 Bell series ： ウィキペディア英語版 Bell series In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell. Given an arithmetic function $f$ and a prime $p$, define the formal power series $f\_p(x)$, called the Bell series of $f$ modulo $p$ as: :$f\_p(x)=\backslash sum\_^\backslash infty\; f(p^n)x^n.$ Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the ''uniqueness theorem'': given multiplicative functions $f$ and $g$, one has $f=g$ if and only if: :$f\_p(x)=g\_p(x)$ for all primes $p$. Two series may be multiplied (sometimes called the ''multiplication theorem''): For any two arithmetic functions $f$ and $g$, let $h=f$ *g be their Dirichlet convolution. Then for every prime $p$, one has: :$h\_p(x)=f\_p(x)\; g\_p(x).\backslash ,$ In particular, this makes it trivial to find the Bell series of a Dirichlet inverse. If $f$ is completely multiplicative, then formally: :$f\_p(x)=\backslash frac.$ ==Examples== The following is a table of the Bell series of well-known arithmetic functions. * The Möbius function $\backslash mu$ has $\backslash mu\_p(x)=1-x.$ * Euler's Totient $\backslash varphi$ has $\backslash varphi\_p(x)=\backslash frac.$ * The multiplicative identity of the Dirichlet convolution $\backslash delta$ has $\backslash delta\_p(x)=1.$ * The Liouville function $\backslash lambda$ has $\backslash lambda\_p(x)=\backslash frac.$ * The power function Idk has $(\backslash textrm\_k)\_p(x)=\backslash frac.$ Here, Idk is the completely multiplicative function $\backslash operatorname\_k(n)=n^k$. * The divisor function $\backslash sigma\_k$ has $(\backslash sigma\_k)\_p(x)=\backslash frac.$ 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「Bell series」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.