Fish curve - Revision history

en>Tewapack at 19:55, 7 May 2014

2014-05-07T19:55:42Z

← Older revision		Revision as of 21:55, 7 May 2014
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	~~{{expert-subject\|Mathematics\|date=November 2009}}~~		Oscar is what my wife enjoys to call me and I completely dig that title. He utilized to be unemployed but now he is a meter reader. North Dakota is exactly where me and my husband live. It's not a common factor but what she likes doing is base leaping and now she is attempting to make money with it.<br><br>my page std testing at home ([http://mtomtech.co.kr/bbs/?document_srl=92816 click to read])
	~~In [[mathematics]], in the area of [[complex analysis]], the '''general difference polynomials''' are a [[polynomial sequence]],~~ a ~~certain subclass of the [[Sheffer polynomials]], which include the [[Newton polynomial]]s, '''Selberg's polynomials''', and the '''Stirling interpolation polynomials''' as special cases~~.

	~~==Definition==~~
	~~The general difference polynomial sequence~~ is ~~given by~~

	~~:<math>p_n(z)=\frac{z}{n} {{z-\beta n -1} \choose {n-1}}</math>~~

	where ~~<math>{z \choose n}</math> is the [[binomial coefficient]]. For <math>\beta=0</math>, the generated polynomials <math>p_n(z)</math> are the Newton polynomials~~

	~~:<math>p_n(z)= {z \choose n} = \frac{z(z-1)\cdots(z-n+1)}{n!}~~.~~</math>~~

	~~The case of <math>\beta=1</math> generates Selberg~~'s ~~polynomials,~~ and ~~the case of <math>\beta=-1/2</math> generates Stirling's interpolation polynomials.~~

	~~==Moving differences==~~
	~~Given an [[analytic function]] <math>f(z)</math>, define the '''moving difference''' of ''f'' as~~

	~~:<math>\mathcal{L}_n(f) = \Delta^n f (\beta n)</math>~~

	~~where <math>\Delta</math>~~ is ~~the [[forward difference operator]]. Then, provided that ''f'' obeys certain summability conditions, then~~ it ~~may be represented in terms of these polynomials as~~

	~~:<math>f(z)=\sum_{n=0}^\infty p_n(z) \mathcal{L}_n(f)~~.<~~/math~~>

	The conditions for summability (that is, convergence) for this sequence is a fairly complex topic; in general, one may say that a necessary condition is that the analytic function be of less than [[exponential type]]. Summability conditions are discussed in detail in Boas & Buck.

	~~==Generating function==~~
	~~The [[generating function]] for the general difference polynomials is given by~~

	:<~~math~~>~~e^{zt}=\sum_{n=0}^\infty p_n(z)~~
	~~\left[\left~~(~~e^t-1\right)e^{\beta t}\right]^n.</math>~~

	~~This generating function can be brought into the form of the [~~[~~generalized Appell representation]]~~

	:~~<math>K(z,w) = A(w)\Psi(zg(w)) = \sum_{n=0}^\infty p_n(z) w^n</math>~~

	~~by setting <math>A(w)=1<~~/~~math>, <math>\Psi(x)=e^x<~~/~~math>, <math>g(w)=t<~~/~~math> and <math>w=(e^t-1)e^{\beta t}<~~/~~math>.~~

	=~~=See also==~~
	* [[Carlson's theorem]]

	~~==References==~~
	~~{{reflist}}~~
	* Ralph P. Boas, Jr. and R. Creighton Buck, ''Polynomial Expansions of Analytic Functions (Second Printing Corrected)~~'', (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263.~~

	~~[[Category:Polynomials]]~~
	~~[[Category:Finite differences]]~~
	~~[[Category:Factorial and binomial topics]]~~

74.111.243.245: /* Equations */

2013-03-18T03:55:52Z

Equations

← Older revision		Revision as of 05:55, 18 March 2013
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	~~Greetings! I am Myrtle Shroyer.~~ In ~~her professional life she is~~ a ~~payroll clerk but she~~'s ~~always needed her own company~~. The ~~factor she adores most~~ is ~~body building and now she~~ is ~~trying to make cash with it~~. ~~My family members life in Minnesota and my family loves it~~.<br><br>~~My web~~-~~site~~ [~~http~~://~~www~~.~~animecontent~~.~~com~~/~~blog~~/~~124234 over~~ the ~~counter std test~~]		{{expert-subject\|Mathematics\|date=November 2009}}
			In [[mathematics]], in the area of [[complex analysis]], the '''general difference polynomials''' are a [[polynomial sequence]], a certain subclass of the [[Sheffer polynomials]], which include the [[Newton polynomial]]s, '''Selberg's polynomials''', and the '''Stirling interpolation polynomials''' as special cases.

			==Definition==
			The general difference polynomial sequence is given by

			:<math>p_n(z)=\frac{z}{n} {{z-\beta n -1} \choose {n-1}}</math>

			where <math>{z \choose n}</math> is the [[binomial coefficient]]. For <math>\beta=0</math>, the generated polynomials <math>p_n(z)</math> are the Newton polynomials

			:<math>p_n(z)= {z \choose n} = \frac{z(z-1)\cdots(z-n+1)}{n!}.</math>

			The case of <math>\beta=1</math> generates Selberg's polynomials, and the case of <math>\beta=-1/2</math> generates Stirling's interpolation polynomials.

			==Moving differences==
			Given an [[analytic function]] <math>f(z)</math>, define the '''moving difference''' of ''f'' as

			:<math>\mathcal{L}_n(f) = \Delta^n f (\beta n)</math>

			where <math>\Delta</math> is the [[forward difference operator]]. Then, provided that ''f'' obeys certain summability conditions, then it may be represented in terms of these polynomials as

			:<math>f(z)=\sum_{n=0}^\infty p_n(z) \mathcal{L}_n(f).</math>

			The conditions for summability (that is, convergence) for this sequence is a fairly complex topic; in general, one may say that a necessary condition is that the analytic function be of less than [[exponential type]]. Summability conditions are discussed in detail in Boas & Buck.

			==Generating function==
			The [[generating function]] for the general difference polynomials is given by

			:<math>e^{zt}=\sum_{n=0}^\infty p_n(z)
			\left[\left(e^t-1\right)e^{\beta t}\right]^n.</math>

			This generating function can be brought into the form of the [[generalized Appell representation]]

			:<math>K(z,w) = A(w)\Psi(zg(w)) = \sum_{n=0}^\infty p_n(z) w^n</math>

			by setting <math>A(w)=1</math>, <math>\Psi(x)=e^x</math>, <math>g(w)=t</math> and <math>w=(e^t-1)e^{\beta t}</math>.

			==See also==
			* [[Carlson's theorem]]

			==References==
			{{reflist}}
			* Ralph P. Boas, Jr. and R. Creighton Buck, ''Polynomial Expansions of Analytic Functions (Second Printing Corrected)'', (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263.

			[[Category:Polynomials]]
			[[Category:Finite differences]]
			[[Category:Factorial and binomial topics]]

en>Krishnavedala: Reverted 1 edit by 117.225.74.182 (talk) identified as vandalism to last revision by Krishnavedala. (TW)

2012-01-31T23:09:54Z

Reverted 1 edit by 117.225.74.182 (talk) identified as vandalism to last revision by Krishnavedala. (TW)

New page

Greetings! I am Myrtle Shroyer. In her professional life she is a payroll clerk but she's always needed her own company. The factor she adores most is body building and now she is trying to make cash with it. My family members life in Minnesota and my family loves it.<br><br>My web-site [http://www.animecontent.com/blog/124234 over the counter std test]