Formula:KLS:09.08:02: Difference between revisions

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<div id="alignleft"> << [[Formula:KLS:09.08:01|Formula:KLS:09.08:01]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:01|Formula:KLS:09.08:01]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:02|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:02|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:03|Formula:KLS:09.08:03]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:03|Formula:KLS:09.08:03]] >> </div>
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<br /><div align="center"><math>{\displaystyle  
<br /><div align="center"><math>{\displaystyle  
\int_{-1}^1(1-x)^{\alpha}(1+x)^{\beta}\Jacobi{\alpha}{\beta}{m}@{x}\Jacobi{\alpha}{\beta}{n}@{x}\,dx
\Ultra{\lambda}{n}@{x}=\frac{\pochhammer{2\lambda}{n}}{n!}\,\HyperpFq{2}{1}@@{-n,n+2\lambda}{\lambda+\frac{1}{2}}{\frac{1-x}{2}}
{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\EulerGamma@{n+\alpha+1}\EulerGamma@{n+\beta+1}}{\EulerGamma@{n+\alpha+\beta+1}n!}\,\Kronecker{m}{n}
}</math></div>
}</math></div>


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== Symbols List ==
== Symbols List ==


<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.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/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/5.2#iii <math>{\displaystyle (a)_n}</math>]</span> : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]<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/16.2#E1 <math>{\displaystyle {{}_{p}F_{q}}}</math>]</span> : generalized hypergeometric function : [http://dlmf.nist.gov/16.2#E1 http://dlmf.nist.gov/16.2#E1]
<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]
<br />
<br />


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<br /><div id="drmf_foot">
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<div id="alignleft"> << [[Formula:KLS:09.08:01|Formula:KLS:09.08:01]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:01|Formula:KLS:09.08:01]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:02|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:02|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:03|Formula:KLS:09.08:03]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:03|Formula:KLS:09.08:03]] >> </div>
</div>
</div>

Revision as of 00:34, 6 March 2017


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{x}=\frac{\pochhammer{2\lambda}{n}}{n!}\,\HyperpFq{2}{1}@@{-n,n+2\lambda}{\lambda+\frac{1}{2}}{\frac{1-x}{2}} }}

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 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 (a)_n}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {{}_{p}F_{q}}}}  : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1

Bibliography

Equation in Section 9.8 of KLS.

URL links

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