Dyck graph - Revision history

2A02:1810:9533:5F00:C37:84D1:40A8:7989: Correct reference on genus

2014-11-18T16:56:52Z

Correct reference on genus

← Older revision		Revision as of 18:56, 18 November 2014
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	~~In mathematics,~~ a ~~'''Neumann polynomial''', introduced~~ by ~~[[Carl Neumann]] for~~ the ~~special case <math>\alpha=0</math>~~, ~~is a polynomial in 1/''z'' used~~ to ~~expand functions in term of [[Bessel function]]s.<ref>Abramowitz~~ and ~~Stegun, [http://www.math.sfu~~.~~ca/~cbm/aands/page_363.htm p. 363~~, ~~9.1.82] ff~~.<~~/ref~~>		Surely the second option would be more beneficial for any website. You may discover this probably the most time-consuming part of building a Word - Press MLM website. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. s ultimately easy to implement and virtually maintenance free. After activating, you will find their website link and get the activation code from their website. <br><br>

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	~~:<math>O_0^{(\alpha)}(t)=\frac 1 t~~,~~</math>~~
	~~:<math>O_1^{(\alpha)}(t)=2\frac {\alpha+1}{t^2},</math>~~
	~~:<math>O_2^{(\alpha)}(t)=\frac {2+\alpha}{t}+ 4\frac {(2+\alpha)(1+\alpha)}{t^3},</math>~~
	~~:<math>O_3^{(\alpha)}(t)=2\frac {(1+\alpha)(3+\alpha)}{t^2}+ 8\frac {(1+\alpha)(2+\alpha)(3+\alpha)}{t^4},</math>~~
	~~:<math>O_4^{(\alpha)}(t)=\frac {(1+\alpha)(4+\alpha)}{2t}+ 4\frac {(1+\alpha)(2+\alpha)(4+\alpha)}{t^3}+ 16\frac {(1+\alpha)(2+\alpha)(3+\alpha)(4+\alpha)}{t^5}~~.~~</math>~~

	A ~~general form for the polynomial~~ is
	~~:<math>O_n^{(\alpha)}(t)= \frac{\alpha+n}{2\alpha} \sum_{k=0}^{\lfloor n/2\rfloor} (-1)^{n-k}\frac {(n-k)!} {k!} {-\alpha \choose n-k}\left(\frac 2 t \right)^{n+1-2k},</math>~~

	~~they have the generating function~~
	~~:<math>\frac{\left(\frac z 2 \right)^\alpha} {\Gamma(\alpha+1)} \frac 1 {t-z}= \sum_{n=0}O_n^{(\alpha)}(t) J_{\alpha+n}(z),</math>~~
	~~where ''J~~'~~' are [[Bessel function]]~~s.

	~~To expand~~ a ~~function ''f'' in form~~
	~~:<math>f(z)=\sum_{n=0} a_n J_{\alpha+n}(z)\,</math>~~
	~~for <math>\|z\|<c</math>~~
	~~compute~~
	~~:<math>a_n=\frac 1 {2 \pi i} \oint_{\|z\|=c'} \frac{\Gamma(\alpha+1)}{\left(\frac z 2\right)^\alpha}f(z) O_n^{(\alpha)}(z)\mathrm d z,</math>~~
	~~where <math>c'<c </math> and ''c''~~ is the ~~distance of the nearest singularity~~ of <~~math~~>~~z^{-\alpha} f(z)~~<~~/math> from <math>z=0</math~~>.

	~~==Examples==~~
	~~An example is the extension~~
	:~~<math>\left(\tfrac{1}{2}z\right)^s= \Gamma(s)\cdot\sum_{k=0}(-1)^k J_{s+2k}(z)(s+2k){-s \choose k}<~~/~~math>~~
	~~or the more general Sonine formula<ref>{{harvnb\|Erdélyi\|Magnus\|Oberhettinger\|Tricomi\|1955}} II.7.10.1, p.64<~~/~~ref>~~
	~~:<math>e^{i \gamma z}= \Gamma(s)\cdot\sum_{k=0}i^k C_k^{(s)}(\gamma)(s+k)\frac{J_{s+k}(z)}{\left(\frac z 2\right)^s}~~.</~~math>~~
	~~where <math>C_k^{(s)}</math> is [[Gegenbauer polynomial\|Gegenbauer's polynomial]~~]. ~~Then~~,~~{{fact\|date=September 2011}}{{or\|date=September 2011}}~~
	~~:<math>\frac{\left(\frac z 2\right)^{2k}}{(2k~~-~~1)!}J_s(z)= \sum_{i=k}(~~-~~1)^{i~~-~~k}{i+k-1\choose 2k-1}{i+k+s-1\choose 2k-1}(s+2i)J_{s+2i}(z),</math>~~
	:<math>\sum_{n=0} t^n J_{s+n}(z)= \frac{e^{\frac{t z}2}}{t^s} \sum_{j=0}\frac{\left(-\frac{z}{2t}\right)^j}{j!}\frac{\gamma \left(j+s,\frac{t z}{2}\right)}{\,\Gamma (j+s)}= \int_0^\infty e^{-\frac{z x^2}{2 t}}\frac {z x}{t} \frac{J_s(z\sqrt{1-x^2})}{\sqrt{1-x^2}^s}\,dx,</math>
	the ~~[[confluent hypergeometric function]]~~
	~~:<math>M(~~a~~,s,z)= \Gamma (s) \sum_{k=0}^\infty \left(-\frac{1}{t}\right)^k L_k^{(~~-a~~-k)}(t) \frac{J_{s+k-1}\left(2 \sqrt{t z}\right)}{(\sqrt{t z})^{s-k-1}}~~<~~/math~~>
	~~and in particular~~
	:<~~math~~>~~\frac{J_s(2 z)}{z^s}= \frac{4^s \Gamma\left(s+\frac12\right)}{\sqrt\pi}e^{2 i z}\sum_{k=0}L_k^{(~~-~~s-1/2-k)}\left(\frac{it}4\right)(4 i z)^k \frac{J_{2s+k}\left(2\sqrt{t z}\right)}{\sqrt{t z}^{2s+k}},</math>~~
	the ~~index shift formula~~
	~~:<math>\Gamma(\nu~~-~~\mu) J_\nu~~(z)~~= \Gamma(\mu+1) \sum_{n=0}\frac{\Gamma(\nu-\mu+n)}{n!\Gamma(\nu+n+1)} \left(\frac z 2\right)^{\nu-\mu+n}J_{\mu+n}(z),</math>~~
	~~the Taylor expansion (addition formula)~~
	:~~<math>\frac{J_s\left(\sqrt{z^2-2uz}\right)}{\left(\sqrt{z^2-2uz}\right)^{\pm s}}= \sum_{k=0}\frac{(\pm u)^k}{k!}\frac{J_{s\pm k}(z)}{z^{\pm s}}</math>~~
	~~(cf.<ref>~~
	I.~~S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжи); Alan Jeffrey, Daniel Zwillinger, editors. '~~'~~Table~~ of ~~Integrals, Series~~, and ~~Products''~~, ~~seventh edition~~. ~~Academic~~ Press~~, 2007~~. ~~ISBN 978~~-0-12-~~373637~~-6. ~~Equation 8.515.1</ref>{{verification failed\|date=September 2011}}<!~~-~~-The referenced identity seems to~~ be ~~only superficially similar. It does~~ not ~~directly support the identity here~~.~~-->)~~ and the ~~expansion of~~ the ~~integral~~ of ~~the Bessel function~~
	~~:<math>\int J_s(z)dz= 2 \sum_{k=0} J_{~~s~~+2k+1}(z)</math>~~
	~~are of the same type~~.

	~~==See also==~~
	*[[Bessel function]]
	*[[Lommel polynomial]]
	*[[Hankel transform]]
	*[[Fourier–Bessel series]]

	~~== Notes ==~~
	~~<references/>~~

	~~[[Category:Polynomials]]~~
	~~[[Category:Special functions]]~~

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3115480

2013-03-11T04:32:00Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3115480

← Older revision		Revision as of 06:32, 11 March 2013
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	I'~~m Matilda and I live~~ in ~~a seaside city~~ in ~~northern France~~, ~~Charleville~~-~~Mezieres~~. I'~~m 21~~ and I'~~m will soon finish my study at Gender and Women~~'s ~~Studies~~.<br><br>~~Also visit my web-site .~~.. [~~http~~://~~www~~.~~londonkba~~.ca/~~?document_srl~~=~~10796576 индивидуальная диета бесплатно~~]		In mathematics, a '''Neumann polynomial''', introduced by [[Carl Neumann]] for the special case <math>\alpha=0</math>, is a polynomial in 1/''z'' used to expand functions in term of [[Bessel function]]s.<ref>Abramowitz and Stegun, [http://www.math.sfu.ca/~cbm/aands/page_363.htm p. 363, 9.1.82] ff.</ref>

			The first few polynomials are
			:<math>O_0^{(\alpha)}(t)=\frac 1 t,</math>
			:<math>O_1^{(\alpha)}(t)=2\frac {\alpha+1}{t^2},</math>
			:<math>O_2^{(\alpha)}(t)=\frac {2+\alpha}{t}+ 4\frac {(2+\alpha)(1+\alpha)}{t^3},</math>
			:<math>O_3^{(\alpha)}(t)=2\frac {(1+\alpha)(3+\alpha)}{t^2}+ 8\frac {(1+\alpha)(2+\alpha)(3+\alpha)}{t^4},</math>
			:<math>O_4^{(\alpha)}(t)=\frac {(1+\alpha)(4+\alpha)}{2t}+ 4\frac {(1+\alpha)(2+\alpha)(4+\alpha)}{t^3}+ 16\frac {(1+\alpha)(2+\alpha)(3+\alpha)(4+\alpha)}{t^5}.</math>

			A general form for the polynomial is
			:<math>O_n^{(\alpha)}(t)= \frac{\alpha+n}{2\alpha} \sum_{k=0}^{\lfloor n/2\rfloor} (-1)^{n-k}\frac {(n-k)!} {k!} {-\alpha \choose n-k}\left(\frac 2 t \right)^{n+1-2k},</math>

			they have the generating function
			:<math>\frac{\left(\frac z 2 \right)^\alpha} {\Gamma(\alpha+1)} \frac 1 {t-z}= \sum_{n=0}O_n^{(\alpha)}(t) J_{\alpha+n}(z),</math>
			where ''J'' are [[Bessel function]]s.

			To expand a function ''f'' in form
			:<math>f(z)=\sum_{n=0} a_n J_{\alpha+n}(z)\,</math>
			for <math>\|z\|<c</math>
			compute
			:<math>a_n=\frac 1 {2 \pi i} \oint_{\|z\|=c'} \frac{\Gamma(\alpha+1)}{\left(\frac z 2\right)^\alpha}f(z) O_n^{(\alpha)}(z)\mathrm d z,</math>
			where <math>c'<c </math> and ''c'' is the distance of the nearest singularity of <math>z^{-\alpha} f(z)</math> from <math>z=0</math>.

			==Examples==
			An example is the extension
			:<math>\left(\tfrac{1}{2}z\right)^s= \Gamma(s)\cdot\sum_{k=0}(-1)^k J_{s+2k}(z)(s+2k){-s \choose k}</math>
			or the more general Sonine formula<ref>{{harvnb\|Erdélyi\|Magnus\|Oberhettinger\|Tricomi\|1955}} II.7.10.1, p.64</ref>
			:<math>e^{i \gamma z}= \Gamma(s)\cdot\sum_{k=0}i^k C_k^{(s)}(\gamma)(s+k)\frac{J_{s+k}(z)}{\left(\frac z 2\right)^s}.</math>
			where <math>C_k^{(s)}</math> is [[Gegenbauer polynomial\|Gegenbauer's polynomial]]. Then,{{fact\|date=September 2011}}{{or\|date=September 2011}}
			:<math>\frac{\left(\frac z 2\right)^{2k}}{(2k-1)!}J_s(z)= \sum_{i=k}(-1)^{i-k}{i+k-1\choose 2k-1}{i+k+s-1\choose 2k-1}(s+2i)J_{s+2i}(z),</math>
			:<math>\sum_{n=0} t^n J_{s+n}(z)= \frac{e^{\frac{t z}2}}{t^s} \sum_{j=0}\frac{\left(-\frac{z}{2t}\right)^j}{j!}\frac{\gamma \left(j+s,\frac{t z}{2}\right)}{\,\Gamma (j+s)}= \int_0^\infty e^{-\frac{z x^2}{2 t}}\frac {z x}{t} \frac{J_s(z\sqrt{1-x^2})}{\sqrt{1-x^2}^s}\,dx,</math>
			the [[confluent hypergeometric function]]
			:<math>M(a,s,z)= \Gamma (s) \sum_{k=0}^\infty \left(-\frac{1}{t}\right)^k L_k^{(-a-k)}(t) \frac{J_{s+k-1}\left(2 \sqrt{t z}\right)}{(\sqrt{t z})^{s-k-1}}</math>
			and in particular
			:<math>\frac{J_s(2 z)}{z^s}= \frac{4^s \Gamma\left(s+\frac12\right)}{\sqrt\pi}e^{2 i z}\sum_{k=0}L_k^{(-s-1/2-k)}\left(\frac{it}4\right)(4 i z)^k \frac{J_{2s+k}\left(2\sqrt{t z}\right)}{\sqrt{t z}^{2s+k}},</math>
			the index shift formula
			:<math>\Gamma(\nu-\mu) J_\nu(z)= \Gamma(\mu+1) \sum_{n=0}\frac{\Gamma(\nu-\mu+n)}{n!\Gamma(\nu+n+1)} \left(\frac z 2\right)^{\nu-\mu+n}J_{\mu+n}(z),</math>
			the Taylor expansion (addition formula)
			:<math>\frac{J_s\left(\sqrt{z^2-2uz}\right)}{\left(\sqrt{z^2-2uz}\right)^{\pm s}}= \sum_{k=0}\frac{(\pm u)^k}{k!}\frac{J_{s\pm k}(z)}{z^{\pm s}}</math>
			(cf.<ref>
			I.S. Gradshteyn (И.С. Градштейн), I.M. Ryzhik (И.М. Рыжи); Alan Jeffrey, Daniel Zwillinger, editors. ''Table of Integrals, Series, and Products'', seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Equation 8.515.1</ref>{{verification failed\|date=September 2011}}<!--The referenced identity seems to be only superficially similar. It does not directly support the identity here.-->) and the expansion of the integral of the Bessel function
			:<math>\int J_s(z)dz= 2 \sum_{k=0} J_{s+2k+1}(z)</math>
			are of the same type.

			==See also==
			*[[Bessel function]]
			*[[Lommel polynomial]]
			*[[Hankel transform]]
			*[[Fourier–Bessel series]]

			== Notes ==
			<references/>

			[[Category:Polynomials]]
			[[Category:Special functions]]

en>David Eppstein: /* Dyck map */ regular map

2010-06-06T07:44:03Z

Dyck map: regular map

New page

I'm Matilda and I live in a seaside city in northern France, Charleville-Mezieres. I'm 21 and I'm will soon finish my study at Gender and Women's Studies.<br><br>Also visit my web-site ... [http://www.londonkba.ca/?document_srl=10796576 индивидуальная диета бесплатно]