Formula:KLS:09.08:17: Difference between revisions

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<div id="drmf_head">
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<div id="alignleft"> << [[Formula:KLS:09.08:16|Formula:KLS:09.08:16]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:16|Formula:KLS:09.08:16]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:17|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:17|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:18|Formula:KLS:09.08:18]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:18|Formula:KLS:09.08:18]] >> </div>
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<br /><div align="center"><math>{\displaystyle  
<br /><div align="center"><math>{\displaystyle  
\lim_{t\rightarrow\infty}
(1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}}
\frac{\Wilson{n}@{\frac{1}{2}(1-x)t^2}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\beta+1)+\iunit t}{\frac{1}{2}(\beta+1}-\iunit t)}
{}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{\pochhammer{2\lambda}{n}}\Ultra{\lambda}{n}@{x}t^n
{t^{2n}n!}=\Jacobi{\alpha}{\beta}{n}@{x}
}</math></div>
}</math></div>
== Constraint(s) ==
<div align="left"><math>{\displaystyle \gamma}</math> arbitrary</div><br />


== Proof ==
== Proof ==
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== Symbols List ==
== Symbols List ==


<span class="plainlinks">[http://dlmf.nist.gov/18.25#T1.t1.r2 <math>{\displaystyle W_{n}}</math>]</span> : Wilson polynomial : [http://dlmf.nist.gov/18.25#T1.t1.r2 http://dlmf.nist.gov/18.25#T1.t1.r2]<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]<br />
<span class="plainlinks">[http://dlmf.nist.gov/1.9.i <math>{\displaystyle \mathrm{i}}</math>]</span> : imaginary unit : [http://dlmf.nist.gov/1.9.i http://dlmf.nist.gov/1.9.i]<br />
<span class="plainlinks">[http://drmf.wmflabs.org/wiki/Definition:sum <math>{\displaystyle \Sigma}</math>]</span> : sum : [http://drmf.wmflabs.org/wiki/Definition:sum http://drmf.wmflabs.org/wiki/Definition:sum]<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]
<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/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 />
<br />


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<br /><div id="drmf_foot">
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<div id="alignleft"> << [[Formula:KLS:09.08:16|Formula:KLS:09.08:16]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:16|Formula:KLS:09.08:16]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:17|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:17|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:18|Formula:KLS:09.08:18]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:18|Formula:KLS:09.08:18]] >> </div>
</div>
</div>

Latest revision as of 07:35, 22 December 2019


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{\pochhammer{2\lambda}{n}}\Ultra{\lambda}{n}@{x}t^n }}

Constraint(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary


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 {{}_{p}F_{q}}}}  : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Sigma}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
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 C^{\mu}_{n}}}  : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5

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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