Formula:KLS:09.08:03: Difference between revisions

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<div id="drmf_head">
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<div id="alignleft"> << [[Formula:KLS:09.08:02|Formula:KLS:09.08:02]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:02|Formula:KLS:09.08:02]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:03|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:03|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:04|Formula:KLS:09.08:04]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:04|Formula:KLS:09.08:04]] >> </div>
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<br /><div align="center"><math>{\displaystyle  
<br /><div align="center"><math>{\displaystyle  
\int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}\Jacobi{\alpha}{\beta}{m}@{-x}\Jacobi{\alpha}{\beta}{n}@{-x}\,dx
\int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}\Ultra{\lambda}{m}@{x}\Ultra{\lambda}{n}@{x}\,dx
{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\EulerGamma@{-n-\alpha-\beta}\EulerGamma@{n+\alpha+\beta+1}}{\EulerGamma@{-n-\alpha}n!}\,\Kronecker{m}{n}
{}=\frac{\cpi\EulerGamma@{n+2\lambda}2^{1-2\lambda}}{\left\{\EulerGamma@{\lambda}\right\}^2(n+\lambda)n!}\,\Kronecker{m}{n}
}</math></div>
}</math></div>
== Constraint(s) ==
<div align="left"><math>{\displaystyle \lambda>-\frac{1}{2}\quad\lambda\neq 0}</math></div><br />


== Proof ==
== Proof ==
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<span class="plainlinks">[http://dlmf.nist.gov/1.4#iv <math>{\displaystyle \int}</math>]</span> : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]<br />
<span class="plainlinks">[http://dlmf.nist.gov/1.4#iv <math>{\displaystyle \int}</math>]</span> : integral : [http://dlmf.nist.gov/1.4#iv http://dlmf.nist.gov/1.4#iv]<br />
<span class="plainlinks">[http://dlmf.nist.gov/18.3#T1.t1.r3 <math>{\displaystyle P^{(\alpha,\beta)}_{n}}</math>]</span> : Jacobi polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r3 http://dlmf.nist.gov/18.3#T1.t1.r3]<br />
<span class="plainlinks">[http://dlmf.nist.gov/18.3#T1.t1.r5 <math>{\displaystyle C^{\mu}_{n}}</math>]</span> : ultraspherical/Gegenbauer polynomial : [http://dlmf.nist.gov/18.3#T1.t1.r5 http://dlmf.nist.gov/18.3#T1.t1.r5]<br />
<span class="plainlinks">[http://dlmf.nist.gov/5.19.E4 <math>{\displaystyle \pi}</math>]</span> : ratio of a circle's circumference to its diameter : [http://dlmf.nist.gov/5.19.E4 http://dlmf.nist.gov/5.19.E4]<br />
<span class="plainlinks">[http://dlmf.nist.gov/5.2#E1 <math>{\displaystyle \Gamma}</math>]</span> : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]<br />
<span class="plainlinks">[http://dlmf.nist.gov/5.2#E1 <math>{\displaystyle \Gamma}</math>]</span> : Euler's gamma function : [http://dlmf.nist.gov/5.2#E1 http://dlmf.nist.gov/5.2#E1]<br />
<span class="plainlinks">[http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 <math>{\displaystyle \delta_{m,n}}</math>]</span> : Kronecker delta : [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4]
<span class="plainlinks">[http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 <math>{\displaystyle \delta_{m,n}}</math>]</span> : Kronecker delta : [http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4 http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4]
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<br /><div id="drmf_foot">
<br /><div id="drmf_foot">
<div id="alignleft"> << [[Formula:KLS:09.08:02|Formula:KLS:09.08:02]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:02|Formula:KLS:09.08:02]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:03|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:03|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:04|Formula:KLS:09.08:04]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:04|Formula:KLS:09.08:04]] >> </div>
</div>
</div>

Latest revision as of 07:34, 22 December 2019


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}\Ultra{\lambda}{m}@{x}\Ultra{\lambda}{n}@{x}\,dx {}=\frac{\cpi\EulerGamma@{n+2\lambda}2^{1-2\lambda}}{\left\{\EulerGamma@{\lambda}\right\}^2(n+\lambda)n!}\,\Kronecker{m}{n} }}

Constraint(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda>-\frac{1}{2}\quad\lambda\neq 0}}


Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int}}  : integral : http://dlmf.nist.gov/1.4#iv
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C^{\mu}_{n}}}  : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \pi}}  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Gamma}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \delta_{m,n}}}  : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.8 of KLS.

URL links

We ask users to provide relevant URL links in this space.


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