Meixner polynomials - Revision history

en>GregorB: Clean up duplicate template arguments using findargdups

2015-01-01T23:14:08Z

Clean up duplicate template arguments using findargdups

← Older revision		Revision as of 01:14, 2 January 2015
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	~~In mathematics, a '''bilateral hypergeometric series'''~~ is ~~a series Σ''a''<sub>''n''</sub> summed over ''all'' integers ''n'', and such that the ratio~~		My namе is Lucia (23 years old) and my hobbіes are Microscopy and Disc golf.<br><br>My Ьlog post ... [http://Www.Itsadunndealrealty.com/vigrx-plus-safe-stay-healthy-and-eat-good-food-using-this-type-of-straightforward-wholesome-program/ vigrx plus forum francais]

	~~:''a''<sub>''n''</sub>/''a''<sub>''n''+1</sub>~~

	of two terms is a rational function of ''n''. The definition of the [[generalized hypergeometric series]] is similar, except that the terms with negative ''n'' must vanish; the bilateral series will in general have infinite numbers of non-zero terms for both positive and negative ''n''.

	The bilateral hypergeometric series fails to converge for most rational functions, though it can be analytically continued to a function defined for most rational functions. There are several summation formulas giving its values for special values where it does converge.

	~~==Definition==~~
	~~The bilateral hypergeometric series <sub>''p''</sub>H<sub>''p''</sub> is defined by~~
	~~:<math>{}_pH_p(a_1,\ldots,a_p;b_1,\ldots,b_p;z)=~~
	~~{}_pH_p\left(\begin{matrix}a_1&\ldots&a_p\\b_1&\ldots&b_p\\ \end{matrix};z\right)=~~
	~~\sum_{n=-\infty}^\infty~~
	~~\frac{(a_1)_n(a_2)_n\ldots(a_p)_n}{(b_1)_n~~(~~b_2~~)~~_n\ldots(b_p)_n}z^n</math>~~

	~~where~~

	~~:<math>(a)_n=a(a+1)(a+2)\cdots(a+n-1)\,</math>~~

	~~is the [[rising factorial]] or [[Pochhammer symbol]].~~

	~~Usually the variable ''z'' is taken to be 1, in which case it is omitted from the notation.~~
	It is possible to define the series <sub>''p''</sub>H<sub>''q''</sub> with different ''p'' and ''q'' in a similar way, but this either fails to converge or can be reduced to the usual hypergeomtric series by changes of variables.

	~~==Convergence~~ and ~~analytic continuation==~~
	Suppose that none of the variables ''a'' or ''b'' are integers, so that all the terms of the series are finite and non-zero. Then the terms with ''n''<0 diverge if \|''z''\| <1, and the terms with ''n''>0 diverge if \|''z''\| >1, so the series cannot converge unless \|''z''\|=1. When \|''z''\|=1, the series converges if
	~~:<math>\Re(b_1+\cdots b_n -a_1-\cdots - a_n) >1. </math>~~

	~~The bilateral hypergeometric series can be analytically continued to a multivalued meromorphic function of several variables whose singularities~~ are
	~~branch points at ''z'' = 0~~ and ~~''z''=1 and simple poles at ''a''<sub>''i''</sub> = −1, −2,~~.~~.. and ''b''~~<~~sub~~>~~''i''~~<~~/sub~~> ~~= 0, 1, 2, .~~..
	~~This can be done as follows. Suppose that none of the ''a'' or ''b'' variables are integers~~. The terms with ''n'' positive converge for \|''z''\| <1 to a function satisfying an inhomogeneous linear equation with singularities at ''z'' = 0 and ''z''=1, so can be continued to a multivalued function with these points as branch points. Similarly the terms with ''n'' negative converge for \|''z''\| >1 to a function satisfying an inhomogeneous linear equation with singularities at ''z'' = 0 and ''z''=1, so can also be continued to a multivalued function with these points as branch points. The sum of these functions gives the analytic continuation of the bilateral hypergeometric series to all values of ''z'' other than 0 and 1, and satisfies a [~~[linear differential equation]] in ''z'' similar to the hypergeometric differential equation.~~

	~~==Summation formulas==~~
	~~===Dougall's bilateral sum===~~

	:~~<math> {}_2H_2(a,b;c,d;1)= \sum_{-\infty}^\infty\frac{(a)_n(b)_n}{(c)_n(d)_n}= \frac{\Gamma(d)\Gamma(c)\Gamma(1-a)\Gamma(1-b)\Gamma(c+d-a-b-1)}{\Gamma(c-a)\Gamma(c-b)\Gamma(d-a)\Gamma(d-b)} <~~/~~math>~~
	~~{{harv\|Dougall\|1907}}~~

	~~This is sometimes written in the equivalent form~~
	~~:<math>\sum_{n=-\infty}^\infty~~
	~~\frac {\Gamma(a+n) \Gamma(b+n)}{\Gamma(c+n)\Gamma(d+n)} =~~
	~~\frac {\pi^2}{\sin (\pi a) \sin (\pi b)}~~
	~~\frac {\Gamma (c+d-a-b-1)}{\Gamma(c-a) \Gamma(d-a) \Gamma(c-b) \Gamma(d-b)}~~.</~~math>~~

	~~===Bailey's formula===~~
	~~{{harv\|Bailey\|1959}} gave the following generalization of Dougall's formula:~~

	~~:<math> {}_3H_3(a,b, f+1;d,e,f;1)=~~
	~~\sum_{-\infty}^\infty\frac{(a)_n(b)_n(f+1)_n}{(d)_n(e)_n(f)_n}= \lambda\frac{\Gamma(d)\Gamma(e)\Gamma(1~~-~~a)\Gamma(1~~-~~b)\Gamma(d+e~~-a-b-~~2)}{\Gamma(d~~-~~a)\Gamma(d~~-~~b)\Gamma(e~~-~~a)\Gamma(e~~-~~b)} </math>~~
	~~where~~
	~~:<math> \lambda=f^{~~-~~1}\left[(f~~-~~a)(f~~-b)-~~(1+f~~-~~d)(1+f-e)\right].</math>~~

	~~==See also==~~
	*[[basic bilateral hypergeometric series]]

	~~==References==~~
	* {{citation \| authorlink= W. N. Bailey \| last1= Bailey \| first1= W. N. \| title= On the sum of a particular bilateral hypergeometric series <sub>3</sub>''H''<sub>3</sub> \| journal= The Quarterly Journal of Mathematics. Oxford. Second Series \| volume= 10 \| year= 1959 \| pages= 92–94 \| issn= 0033-~~5606 \| doi= 10.1093~~/~~qmath/10.1.92 \| mr= 0107727 }}~~
	* {{citation \| last= Dougall \| first= J. \|authorlink=John Dougall (mathematician)\| title= On Vandermonde's Theorem and Some More General Expansions \| journal= Proc. Edinburgh Math. Soc. \| volume= 25 \| year= 1907 \| pages= 114–132 \| doi= 10.1017/S0013091500033642}}
	* {{citation \| last1= Slater \| first1= Lucy Joan \| authorlink= Lucy Joan Slater \| title= Generalized hypergeometric functions \| location= Cambridge, UK \| publisher= Cambridge University Press \| isbn= 0-521-06483-X \| mr= 0201688 \| year= 1966}} (there is a 2008 paperback with ISBN 978-0-521-09061-2)

	~~[[Category:Hypergeometric functions]~~]

en>R. J. Mathar: /* References */ typos

2013-10-27T19:19:55Z

References: typos

← Older revision		Revision as of 21:19, 27 October 2013
Line 1:		Line 1:
	~~Hello! <br>My name~~ is ~~Johnette and I~~'m a ~~27 years old boy from Wroclaw.~~<br>xunjie 短いスカートスタイルクラッシュOufenカラーシャツ+ハイウエストスカートと着用する女性のためのかわいいフラウンス付きカラーシャツは襟短いスカートスタイルのクラッシュで着用する女性のためのかわいいシャツをフラウンス付き。		In mathematics, a '''bilateral hypergeometric series''' is a series Σ''a''<sub>''n''</sub> summed over ''all'' integers ''n'', and such that the ratio
	~~輸出の製品の品質とグレードを向上させるため、~~
	~~映画やテレビタレント、 [http:~~/~~/www.steadfast-hawaii.org/img/p/hot/chloe.php �� Хå� ��] 専門的なファッション企業の一つとして、~~
	~~それを再利用するのは面倒。~~
	そして涙の影響をインクデザインテーマ、 [http://bigapplefilmfestival.com/css/shop/Duvetica.php ��󥯥�`�� ȥ�å�] 輸入品の関税評価（参考価格）は3,~~400種類を上げることにしました。~~
	~~市場価格のダイナミクスが島フィラメント周盛市場ことを示しています市場中心価格の価格の安定、~~
	愛の愛好家を構成する春の光足音を取る動きが、[http://www.schochauer.ch/_js/r/e/mall/watch/gaga/ ��ߥ�� rӋ ��] ラオス風水翔（1848年に開始したが、
	~~ランバンは盲目的に流行のファッション王国をフォローすることはありません。~~
	~~多くの新聞やポスター、~~
	スヌーカー面白い動画がインターネットユーザークレイジーパスラッシュとコメントの大半を引き起こして、 [http://www.steadfast-hawaii.org/img/p/hot/chloe.php ��<br><br>ؔ�� ]

	~~my weblog~~ ... [~~http~~://~~www~~.~~swissskogkatt~~.ch/~~libs~~/e/~~store~~/~~gaga~~.~~php ガガミラノ腕時計~~]		:''a''<sub>''n''</sub>/''a''<sub>''n''+1</sub>

			of two terms is a rational function of ''n''. The definition of the [[generalized hypergeometric series]] is similar, except that the terms with negative ''n'' must vanish; the bilateral series will in general have infinite numbers of non-zero terms for both positive and negative ''n''.

			The bilateral hypergeometric series fails to converge for most rational functions, though it can be analytically continued to a function defined for most rational functions. There are several summation formulas giving its values for special values where it does converge.

			==Definition==
			The bilateral hypergeometric series <sub>''p''</sub>H<sub>''p''</sub> is defined by
			:<math>{}_pH_p(a_1,\ldots,a_p;b_1,\ldots,b_p;z)=
			{}_pH_p\left(\begin{matrix}a_1&\ldots&a_p\\b_1&\ldots&b_p\\ \end{matrix};z\right)=
			\sum_{n=-\infty}^\infty
			\frac{(a_1)_n(a_2)_n\ldots(a_p)_n}{(b_1)_n(b_2)_n\ldots(b_p)_n}z^n</math>

			where

			:<math>(a)_n=a(a+1)(a+2)\cdots(a+n-1)\,</math>

			is the [[rising factorial]] or [[Pochhammer symbol]].

			Usually the variable ''z'' is taken to be 1, in which case it is omitted from the notation.
			It is possible to define the series <sub>''p''</sub>H<sub>''q''</sub> with different ''p'' and ''q'' in a similar way, but this either fails to converge or can be reduced to the usual hypergeomtric series by changes of variables.

			==Convergence and analytic continuation==
			Suppose that none of the variables ''a'' or ''b'' are integers, so that all the terms of the series are finite and non-zero. Then the terms with ''n''<0 diverge if \|''z''\| <1, and the terms with ''n''>0 diverge if \|''z''\| >1, so the series cannot converge unless \|''z''\|=1. When \|''z''\|=1, the series converges if
			:<math>\Re(b_1+\cdots b_n -a_1-\cdots - a_n) >1. </math>

			The bilateral hypergeometric series can be analytically continued to a multivalued meromorphic function of several variables whose singularities are
			branch points at ''z'' = 0 and ''z''=1 and simple poles at ''a''<sub>''i''</sub> = −1, −2,... and ''b''<sub>''i''</sub> = 0, 1, 2, ...
			This can be done as follows. Suppose that none of the ''a'' or ''b'' variables are integers. The terms with ''n'' positive converge for \|''z''\| <1 to a function satisfying an inhomogeneous linear equation with singularities at ''z'' = 0 and ''z''=1, so can be continued to a multivalued function with these points as branch points. Similarly the terms with ''n'' negative converge for \|''z''\| >1 to a function satisfying an inhomogeneous linear equation with singularities at ''z'' = 0 and ''z''=1, so can also be continued to a multivalued function with these points as branch points. The sum of these functions gives the analytic continuation of the bilateral hypergeometric series to all values of ''z'' other than 0 and 1, and satisfies a [[linear differential equation]] in ''z'' similar to the hypergeometric differential equation.

			==Summation formulas==
			===Dougall's bilateral sum===

			:<math> {}_2H_2(a,b;c,d;1)= \sum_{-\infty}^\infty\frac{(a)_n(b)_n}{(c)_n(d)_n}= \frac{\Gamma(d)\Gamma(c)\Gamma(1-a)\Gamma(1-b)\Gamma(c+d-a-b-1)}{\Gamma(c-a)\Gamma(c-b)\Gamma(d-a)\Gamma(d-b)} </math>
			{{harv\|Dougall\|1907}}

			This is sometimes written in the equivalent form
			:<math>\sum_{n=-\infty}^\infty
			\frac {\Gamma(a+n) \Gamma(b+n)}{\Gamma(c+n)\Gamma(d+n)} =
			\frac {\pi^2}{\sin (\pi a) \sin (\pi b)}
			\frac {\Gamma (c+d-a-b-1)}{\Gamma(c-a) \Gamma(d-a) \Gamma(c-b) \Gamma(d-b)}.</math>

			===Bailey's formula===
			{{harv\|Bailey\|1959}} gave the following generalization of Dougall's formula:

			:<math> {}_3H_3(a,b, f+1;d,e,f;1)=
			\sum_{-\infty}^\infty\frac{(a)_n(b)_n(f+1)_n}{(d)_n(e)_n(f)_n}= \lambda\frac{\Gamma(d)\Gamma(e)\Gamma(1-a)\Gamma(1-b)\Gamma(d+e-a-b-2)}{\Gamma(d-a)\Gamma(d-b)\Gamma(e-a)\Gamma(e-b)} </math>
			where
			:<math> \lambda=f^{-1}\left[(f-a)(f-b)-(1+f-d)(1+f-e)\right].</math>

			==See also==
			*[[basic bilateral hypergeometric series]]

			==References==
			* {{citation \| authorlink= W. N. Bailey \| last1= Bailey \| first1= W. N. \| title= On the sum of a particular bilateral hypergeometric series <sub>3</sub>''H''<sub>3</sub> \| journal= The Quarterly Journal of Mathematics. Oxford. Second Series \| volume= 10 \| year= 1959 \| pages= 92–94 \| issn= 0033-5606 \| doi= 10.1093/qmath/10.1.92 \| mr= 0107727 }}
			* {{citation \| last= Dougall \| first= J. \|authorlink=John Dougall (mathematician)\| title= On Vandermonde's Theorem and Some More General Expansions \| journal= Proc. Edinburgh Math. Soc. \| volume= 25 \| year= 1907 \| pages= 114–132 \| doi= 10.1017/S0013091500033642}}
			* {{citation \| last1= Slater \| first1= Lucy Joan \| authorlink= Lucy Joan Slater \| title= Generalized hypergeometric functions \| location= Cambridge, UK \| publisher= Cambridge University Press \| isbn= 0-521-06483-X \| mr= 0201688 \| year= 1966}} (there is a 2008 paperback with ISBN 978-0-521-09061-2)

			[[Category:Hypergeometric functions]]

en>R.e.b.: rewrite

2011-08-28T00:43:31Z

rewrite

New page

Hello! <br>My name is Johnette and I'm a 27 years old boy from Wroclaw.<br>xunjie 短いスカートスタイルクラッシュOufenカラーシャツ+ハイウエストスカートと着用する女性のためのかわいいフラウンス付きカラーシャツは襟短いスカートスタイルのクラッシュで着用する女性のためのかわいいシャツをフラウンス付き。
輸出の製品の品質とグレードを向上させるため、
映画やテレビタレント、 [http://www.steadfast-hawaii.org/img/p/hot/chloe.php �� Хå� ��] 専門的なファッション企業の一つとして、
それを再利用するのは面倒。
そして涙の影響をインクデザインテーマ、 [http://bigapplefilmfestival.com/css/shop/Duvetica.php ��󥯥�`�� ȥ�å�] 輸入品の関税評価（参考価格）は3,400種類を上げることにしました。
市場価格のダイナミクスが島フィラメント周盛市場ことを示しています市場中心価格の価格の安定、
愛の愛好家を構成する春の光足音を取る動きが、[http://www.schochauer.ch/_js/r/e/mall/watch/gaga/ ��ߥ�� rӋ ��] ラオス風水翔（1848年に開始したが、
ランバンは盲目的に流行のファッション王国をフォローすることはありません。
多くの新聞やポスター、
スヌーカー面白い動画がインターネットユーザークレイジーパスラッシュとコメントの大半を引き起こして、 [http://www.steadfast-hawaii.org/img/p/hot/chloe.php ��<br><br>ؔ�� ]

my weblog ... [http://www.swissskogkatt.ch/libs/e/store/gaga.php ガガミラノ腕時計]