Continued fraction - Revision history

92.232.127.115 at 10:44, 2 January 2015

2015-01-02T10:44:49Z

← Older revision		Revision as of 12:44, 2 January 2015
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en>Arpan Mathur: Changing the way in which 1/4 is written

2014-03-03T12:10:35Z

Changing the way in which 1/4 is written

← Older revision		Revision as of 14:10, 3 March 2014
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	~~{{other uses\|List of things named after Leonhard Euler#Euler—numbers}}~~		Hello! <br>I'm Chinese female :). <br>I really like The Vampire Diaries!<br><br>my blog post: [http://www.cut-shop.com/ แบตตาเลี่ยน]
	~~In [[number theory]], the '''Euler numbers''' are a [[sequence]] ''E~~<~~sub~~>~~n</sub>'~~' ~~of [[integer]]s {{OEIS\|A122045}} defined by the following [[Taylor series]] expansion:~~

	:~~<math>\frac{1}{\cosh t} = \frac{2}{e^{t} + e^ {-t} } = \sum_{n=0}^\infty \frac{E_n}{n!} \cdot t^n\!</math>~~

	~~where cosh ''t'' is the [[Hyperbolic function\|hyperbolic cosine]]. The Euler numbers appear as a special value of the [[Euler polynomials]].~~

	~~The odd-indexed Euler numbers are all [[0 (number~~)~~\|zero]]. The even-indexed ones {{OEIS\|id=A028296}} have alternating signs~~. ~~Some values are:~~
	~~:''E''<sub>0</sub> = 1~~
	~~:''E''<sub>2</sub> = −1~~
	~~:''E''<sub>4</sub> = 5~~
	~~:''E''<sub>6</sub> = −61~~
	~~:''E''<sub>8</sub> = 1,385~~
	~~:''E''<sub>10</sub> = −50,521~~
	~~:''E''<sub>12</sub> = 2,702,765~~
	~~:''E''<sub>14</sub> = −199,360,981~~
	~~:''E''<sub>16</sub> = 19,391,512,145~~
	~~:''E''<sub>18~~<~~/sub~~> ~~= −2,404,879,675,441~~

	~~Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, and/or change all signs to positive. This encyclopedia adheres to the convention adopted above.~~

	The Euler numbers appear in the [[Taylor series]] expansions of the [[trigonometric function\|secant]] and [[hyperbolic secant]] functions. The latter is the function in the definition. They also occur in [[combinatorics]], specifically when counting the number of [[alternating permutation]]s of a set with an even number of elements.

	~~==Explicit formulas==~~

	~~===Iterated sum===~~
	~~An explicit formula for Euler numbers is given by:~~<~~ref~~>[http://www.voofie.com/content/117/an-explicit-formula-for-the-euler-zigzag-numbers-updown-numbers-from-power-series/ Ross Tang, "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series"]{{dead link\|date=October 2013}}<~~/ref~~>

	:~~<math>E_{2n}=i\sum _{k=1}^{2n+1} \sum _{j=0}^k {k\choose j}\frac{(-1)^j(k-2j)^{2n+1}}{2^k i^k k}</math>~~

	~~where ''i'' denotes the [~~[~~imaginary unit]] with ''i''<sup>2</sup>=−1.~~

	~~===Sum over partitions===~~
	The Euler number ''E''<sub>2''n''</sub> can be expressed as a sum over the even [[Partition (number theory)\|partitions]] of 2''n'',<ref>{{cite journal\|first1=David C.\|last1= Vella\|title=Explicit Formulas for Bernoulli and Euler Numbers\|journal=Integers\|volume=8\|issue=1\|pages=A1\|year=2008\|url= http://www.~~integers~~-~~ejcnt~~.~~org/vol8.html}}</ref>~~

	~~:<math> E_{2n} = (2n)! \sum_{0 \leq k_1, \ldots, k_n \leq n}~ \left( \begin{array}{c} K \\ k_1, \ldots , k_n \end{array} \right)~~
	~~\delta_{n,\sum mk_m } \left( \frac{-1~}{2!} \right)^{k_1} \left( \frac{-1~}{4!} \right)^{k_2}~~
	~~\cdots \left( \frac{-1~}{(2n)!} \right)^{k_n} ,<~~/~~math>~~

	as well as a sum over the odd partitions of 2''n'' − 1,<ref>{{cite arxiv\|eprint=1103.1585\|first1= J.\|last1=Malenfant\|title=Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers}}</ref>

	~~:<math> E_{2n} = (-1)^{n-1} (2n-1)! \sum_{0 \leq k_1, \ldots, k_n \leq 2n-1}~~
	~~\left( \begin{array} {c} K \\ k_1, \ldots , k_n \end{array} \right)~~
	~~\delta_{2n-1,\sum (2m-1)k_m } \left( \frac{-1~}{1!} \right)^{k_1} \left( \frac{1}{3!} \right)^{k_2}~~
	~~\cdots \left( \frac{(-1)^n}{(2n-1)!} \right)^{k_n} , </math>~~

	~~where in both cases <math> K =k_1 + \cdots + k_n</math> and~~
	~~:<math> \left( \begin{array}{c} K \\ k_1, \ldots , k_n \end{array} \right)~~
	~~\equiv \frac{ K!}{k_1! \cdots k_n!}</math>~~
	~~is a [[multinomial coefficient]]. The [[Kronecker delta]]'s in the above formulas restrict the sums over the ''k'''s to <math> 2k_1 + 4k_2 + \cdots +2nk_n=2n</math> and to~~
	~~<math> k_1 + 3k_2 + \cdots +(2n-1)k_n=2n-1</math>, respectively.~~

	~~As an example,~~
	~~:<math>~~
	~~\begin{align}~~
	~~E_{10} & = 10! \left( - \frac{1}{10!} + \frac{2}{2!8!} + \frac{2}{4!6!}~~
	~~- \frac{3}{2!^2 6!}- \frac{3}{2!4!^2} +\frac{4}{2!^3 4!} - \frac{1}{2!^5}\right) \\~~
	~~& = 9! \left( - \frac{1}{9!} + \frac{3}{1!^27!} + \frac{6}{1!3!5!}~~
	~~+\frac{1}{3!^3}- \frac{5}{1!^45!} -\frac{10}{1!^33!^2} + \frac{7}{1!^6 3!} - \frac{1}{1!^9}\right) \\~~
	~~& = -50,521.~~
	~~\end{align}~~
	~~</math>~~

	~~===Determinant===~~
	~~''E''<sub>2''n''</sub> is also given by the [[determinant]]~~

	~~:<math>~~
	~~\begin{align}~~
	~~E_{2n} &=(-1)^n (2n)!~ \begin{vmatrix} \frac{1}{2!}& 1 &~& ~&~\\~~
	~~\frac{1}{4!}& \frac{1}{2!} & 1 &~&~\\~~
	~~\vdots & ~ & \ddots~~ &\ddots~~ & ~\\~~
	~~\frac{1}{(2n-2)!}& \frac{1}{(2n-4)!}& ~&\frac{1}{2!} & 1\\~~
	~~\frac{1}{(2n)!}&\frac{1}{(2n-2)!}& \cdots & \frac{1}{4!} & \frac{1}{2!}\end{vmatrix}.~~

	~~\end{align}~~
	~~</math>~~

	~~==Asymptotic approximation==~~

	~~The Euler numbers grow quite rapidly for large indices as~~
	~~they have the following lower bound~~

	~~: <math> \|E_{2 n}\| > 8 \sqrt { \frac{n}{\pi} } \left(\frac{4 n}{ \pi e}\right)^{2 n}. </math>~~

	~~==See also==~~
	* [[Bell number]]
	* [[Bernoulli number]]
	* [[Euler–Mascheroni constant]]

	~~==References==~~
	~~{{Reflist}}~~

	~~==External links==~~
	* {{springer\|title=Euler numbers\|id=p/e036540}}
	* {{MathWorld\|urlname=EulerNumber\|title=Euler number}}

	~~{{DEFAULTSORT:Euler Number}}~~
	~~[[Category:Integer sequences]~~]

en>Monkbot: /* History of continued fractions */Fix CS1 deprecated date parameter errors

2014-02-04T06:50:15Z

History of continued fractions: Fix CS1 deprecated date parameter errors

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en>Anita5192: /* Notations for continued fractions */ Punctuation.

2012-08-11T00:53:48Z

Notations for continued fractions: Punctuation.

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