Formula:KLS:09.08:14: Difference between revisions

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<div id="alignleft"> << [[Formula:KLS:09.08:13|Formula:KLS:09.08:13]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:13|Formula:KLS:09.08:13]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:14|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:14|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:15|Formula:KLS:09.08:15]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:15|Formula:KLS:09.08:15]] >> </div>
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<br /><div align="center"><math>{\displaystyle  
<br /><div align="center"><math>{\displaystyle  
(1-t)^{-\alpha-\beta-1}\,\HyperpFq{2}{1}@@{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2)}{\alpha+1}{\frac{2(x-1)t}{(1-t)^2}}
\HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\
{}=\sum_{n=0}^{\infty}\frac{\pochhammer{\alpha+\beta+1}{n}}{\pochhammer{\alpha+1}{n}}\Jacobi{\alpha}{\beta}{n}@{x}t^n
\HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}}
=\sum_{n=0}^{\infty}\frac{\Ultra{\lambda}{n}@{x}}
{\pochhammer{2\lambda}{n}\pochhammer{\lambda+\frac{1}{2}}{n}}t^n
}</math></div>
}</math></div>


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<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/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://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://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/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 />
<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 />
<br />


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<br /><div id="drmf_foot">
<br /><div id="drmf_foot">
<div id="alignleft"> << [[Formula:KLS:09.08:13|Formula:KLS:09.08:13]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:13|Formula:KLS:09.08:13]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:14|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:14|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:15|Formula:KLS:09.08:15]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:15|Formula:KLS:09.08:15]] >> </div>
</div>
</div>

Revision as of 23:34, 5 March 2017


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

Proof

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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 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

Bibliography

Equation in Section 9.8 of KLS.

URL links

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