Formula:KLS:09.08:16: Difference between revisions
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imported>SeedBot DRMF |
imported>SeedBot DRMF |
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<div id="drmf_head"> | <div id="drmf_head"> | ||
<div id="alignleft"> << [[Formula:KLS:09.08:15|Formula:KLS:09.08:15]] </div> | <div id="alignleft"> << [[Formula:KLS:09.08:15|Formula:KLS:09.08:15]] </div> | ||
<div id="aligncenter"> [[Jacobi#KLS:09.08:16|formula in Jacobi]] </div> | <div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:16|formula in Jacobi: Special cases]] </div> | ||
<div id="alignright"> [[Formula:KLS:09.08:17|Formula:KLS:09.08:17]] >> </div> | <div id="alignright"> [[Formula:KLS:09.08:17|Formula:KLS:09.08:17]] >> </div> | ||
</div> | </div> | ||
<br /><div align="center"><math>{\displaystyle | <br /><div align="center"><math>{\displaystyle | ||
\HyperpFq{2}{1}@@{\gamma,\ | \HyperpFq{2}{1}@@{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\ | ||
\HyperpFq{2}{1}@@{\gamma,\ | \HyperpFq{2}{1}@@{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}} | ||
{}=\sum_{n=0}^{\infty} | {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\pochhammer{2\lambda-\gamma}{n}} | ||
\frac{\pochhammer{\gamma}{n}\pochhammer{\ | {\pochhammer{2\lambda}{n}\pochhammer{\lambda+\frac{1}{2}}{n}}\Ultra{\lambda}{n}@{x}t^n | ||
}</math></div> | }</math></div> | ||
== Substitution(s) == | == Substitution(s) == | ||
<div align="left"><math>{\displaystyle R=\sqrt{1-2xt+t^2}}</math></div><br /> | <div align="left"><math>{\displaystyle R=\sqrt{1-2xt+t^2}}</math> &<br /> <math>{\displaystyle \gamma}</math> arbitrary</div><br /> | ||
== Proof == | == Proof == | ||
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== Symbols List == | == Symbols List == | ||
& : logical and<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/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/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. | <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"> | <br /><div id="drmf_foot"> | ||
<div id="alignleft"> << [[Formula:KLS:09.08:15|Formula:KLS:09.08:15]] </div> | <div id="alignleft"> << [[Formula:KLS:09.08:15|Formula:KLS:09.08:15]] </div> | ||
<div id="aligncenter"> [[Jacobi#KLS:09.08:16|formula in Jacobi]] </div> | <div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:16|formula in Jacobi: Special cases]] </div> | ||
<div id="alignright"> [[Formula:KLS:09.08:17|Formula:KLS:09.08:17]] >> </div> | <div id="alignright"> [[Formula:KLS:09.08:17|Formula:KLS:09.08:17]] >> </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 \HyperpFq{2}{1}@@{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\pochhammer{2\lambda-\gamma}{n}} {\pochhammer{2\lambda}{n}\pochhammer{\lambda+\frac{1}{2}}{n}}\Ultra{\lambda}{n}@{x}t^n }}
Substitution(s)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R=\sqrt{1-2xt+t^2}}}
&
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary
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
& : logical and
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.