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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 Dickson polynomials ： ウィキペディア英語版 Dickson polynomial In mathematics, the Dickson polynomials (or Brewer polynomials), denoted ''D''''n''(''x'',α), form a polynomial sequence introduced by and rediscovered by in his study of Brewer sums. Over the complex numbers, Dickson polynomials are essentially equivalent to Chebyshev polynomials with a change of variable, and in fact Dickson polynomials are sometimes called Chebyshev polynomials. Dickson polynomials are mainly studied over finite fields, when they are not equivalent to Chebyshev polynomials. One of the main reasons for interest in them is that for fixed α, they give many examples of permutation polynomials: polynomials acting as permutations of finite fields. ==Definition== ''D''0(''x'',α) = 2, and for ''n'' > 0 Dickson polynomials (of the first kind) are given by :$D\_n(x,\backslash alpha)=\backslash sum\_^\backslash frac\; \backslash binom\; (-\backslash alpha)^p\; x^.$ The first few Dickson polynomials are :$D\_0(x,\backslash alpha)\; =\; 2\; \backslash ,$ :$D\_1(x,\backslash alpha)\; =\; x\; \backslash ,$ :$D\_2(x,\backslash alpha)\; =\; x^2\; -\; 2\backslash alpha\; \backslash ,$ :$D\_3(x,\backslash alpha)\; =\; x^3\; -\; 3x\backslash alpha\; \backslash ,$ :$D\_4(x,\backslash alpha)\; =\; x^4\; -\; 4x^2\backslash alpha\; +\; 2\backslash alpha^2.\; \backslash ,$ The Dickson polynomials of the second kind ''E''''n'' are defined by :$E\_n(x,\backslash alpha)=\backslash sum\_^\backslash binom\; (-\backslash alpha)^p\; x^.$ They have not been studied much, and have properties similar to those of Dickson polynomials of the first kind. The first few Dickson polynomials of the second kind are :$E\_0(x,\backslash alpha)\; =\; 1\; \backslash ,$ :$E\_1(x,\backslash alpha)\; =\; x\; \backslash ,$ :$E\_2(x,\backslash alpha)\; =\; x^2\; -\; \backslash alpha\; \backslash ,$ :$E\_3(x,\backslash alpha)\; =\; x^3\; -\; 2x\backslash alpha\; \backslash ,$ :$E\_4(x,\backslash alpha)\; =\; x^4\; -\; 3x^2\backslash alpha\; +\; \backslash alpha^2.\; \backslash ,$ 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「Dickson polynomial」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.