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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 Bernoulli number ： ウィキペディア英語版 Bernoulli number |- | 2 || $\backslash frac$ |- style="background:#ABE" | 3 || 0 |- | 4 || $-\backslash frac$ |- style="background:#ABE" | 5 || 0 |- | 6 || $\backslash frac$ |- style="background:#ABE" | 7 || 0 |- | 8 || $-\backslash frac$ |- style="background:#ABE" | 9 || 0 |- | 10 || $\backslash frac$ |- style="background:#ABE" | 11 || 0 |- | 12 || $-\backslash frac$ |- style="background:#ABE" | 13 || 0 |- | 14 || $\backslash frac$ |- style="background:#ABE" | 15 || 0 |- | 16 || $-\backslash frac$ |- style="background:#ABE" | 17 || 0 |- | 18 || $\backslash frac$ |- style="background:#ABE" | 19 || 0 |- | 20 || $-\backslash frac$ |} In mathematics, the Bernoulli numbers ''B''''n'' are a sequence of rational numbers with deep connections to number theory. The values of the first few Bernoulli numbers are : ''B''0 = 1, ''B''1 = ±, ''B''2 = , ''B''3 = 0, ''B''4 = −, ''B''5 = 0, ''B''6 = , ''B''7 = 0, ''B''8 = −. If the convention ''B''1 = − is used, this sequence is also known as the first Bernoulli numbers ( / in OEIS); with the convention ''B''1 = + is known as the second Bernoulli numbers ( / ). Except for this one difference, the first and second Bernoulli numbers agree. Since ''B''''n'' = 0 for all odd ''n'' > 1, and many formulas only involve even-index Bernoulli numbers, some authors write ''B''''n'' instead of ''B''2''n''. The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jakob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Seki's discovery was posthumously published in 1712〔Selin, H. (1997), p. 891〕〔Smith, D. E. (1914), p. 108〕 in his work ''Katsuyo Sampo''; Bernoulli's, also posthumously, in his ''Ars Conjectandi'' of 1713. Ada Lovelace's note G on the analytical engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine.〔''Note G'' in the Menabrea reference〕 As a result, the Bernoulli numbers have the distinction of being the subject of one of the first computer programs. == Sum of powers == (詳細はclosed form expression of the sum of the ''m''-th powers of the first ''n'' positive integers. For ''m'', ''n'' ≥ 0 define :$S\_m(n)\; =\; \backslash sum\_^n\; k^m\; =\; 1^m\; +\; 2^m\; +\; \backslash cdots\; +\; n^m.\; \backslash ,$ This expression can always be rewritten as a polynomial in ''n'' of degree ''m'' + 1. The coefficients of these polynomials are related to the Bernoulli numbers by Bernoulli's formula: : $S\_m(n)\; =\; ^m\; ,$ where the convention ''B''1 = +1/2 is used. ($\backslash tbinom$ denotes the binomial coefficient, ''m''+1 choose ''k''.) For example, taking ''m'' to be 1 gives the triangular numbers 0, 1, 3, 6, ... . :$1\; +\; 2\; +\; \backslash cdots\; +\; n\; =\; \backslash frac\backslash left(B\_0\; n^2+2B\_1\; n^1\backslash right)\; =\; \backslash frac\backslash left(n^2+n\backslash right).$ Taking ''m'' to be 2 gives the square pyramidal numbers 0, 1, 5, 14, ... . : $1^2\; +\; 2^2\; +\; \backslash cdots\; +\; n^2\; =\; \backslash frac\backslash left(B\_0\; n^3+3B\_1\; n^2+3B\_2\; n^1\; \backslash right)\; =\; \backslash frac\backslash left(n^3+\backslash fracn^2+\backslash fracn\backslash right).$ Some authors use the convention ''B''1 = −1/2 and state Bernoulli's formula in this way: : $S\_m(n)\; =\; ^m\; (-1)^k\; .$ Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who also found remarkable ways to calculate sums of powers. Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog . 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「Bernoulli number」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.