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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 Fibonacci polynomials ： ウィキペディア英語版 Fibonacci polynomials In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials. ==Definition== These Fibonacci polynomials are defined by a recurrence relation:〔Benjamin & Quinn p. 141〕 :$F\_n(x)=\; \backslash begin$ 0, & \mbox n = 0\\ 1, & \mbox n = 1\\ x F_(x) + F_(x),& \mbox n \geq 2 \end The first few Fibonacci polynomials are: :$F\_0(x)=0\; \backslash ,$ :$F\_1(x)=1\; \backslash ,$ :$F\_2(x)=x\; \backslash ,$ :$F\_3(x)=x^2+1\; \backslash ,$ :$F\_4(x)=x^3+2x\; \backslash ,$ :$F\_5(x)=x^4+3x^2+1\; \backslash ,$ :$F\_6(x)=x^5+4x^3+3x\; \backslash ,$ The Lucas polynomials use the same recurrence with different starting values:〔Benjamin & Quinn p. 142〕 $L\_n(x)\; =\; \backslash begin$ 2, & \mbox n = 0 \\ x, & \mbox n = 1 \\ x L_(x) + L_(x), & \mbox n \geq 2. \end The first few Lucas polynomials are: :$L\_0(x)=2\; \backslash ,$ :$L\_1(x)=x\; \backslash ,$ :$L\_2(x)=x^2+2\; \backslash ,$ :$L\_3(x)=x^3+3x\; \backslash ,$ :$L\_4(x)=x^4+4x^2+2\; \backslash ,$ :$L\_5(x)=x^5+5x^3+5x\; \backslash ,$ :$L\_6(x)=x^6+6x^4+9x^2\; +\; 2.\; \backslash ,$ The Fibonacci and Lucas numbers are recovered by evaluating the polynomials at ''x'' = 1; Pell numbers are recovered by evaluating ''F''''n'' at ''x'' = 2. The degrees of ''F''''n'' is ''n'' − 1 and the degree of ''L''''n'' is ''n''. The ordinary generating function for the sequences are: :$\backslash sum\_^\backslash infty\; F\_n(x)\; t^n\; =\; \backslash frac$ :$\backslash sum\_^\backslash infty\; L\_n(x)\; t^n\; =\; \backslash frac.$ The polynomials can be expressed in terms of Lucas sequences as :$F\_n(x)\; =\; U\_n(x,-1),\backslash ,$ :$L\_n(x)\; =\; V\_n(x,-1).\backslash ,$ 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「Fibonacci polynomials」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.