Formula:KLS:09.08:13: Difference between revisions

From DRMF
Jump to navigation Jump to search
Newer edit →
imported>SeedBot
DRMF
 
imported>SeedBot
DRMF
Line 2: Line 2:
<div id="drmf_head">
<div id="drmf_head">
<div id="alignleft"> << [[Formula:KLS:09.08:12|Formula:KLS:09.08:12]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:12|Formula:KLS:09.08:12]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:13|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:13|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:14|Formula:KLS:09.08:14]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:14|Formula:KLS:09.08:14]] >> </div>
</div>
</div>


<br /><div align="center"><math>{\displaystyle  
<br /><div align="center"><math>{\displaystyle  
\HyperpFq{0}{1}@@{-}{\alpha+1}{\frac{(x-1)t}{2}}\,\HyperpFq{0}{1}@@{-}{\beta+1}{\frac{(x+1)t}{2}}
R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty}
{}=\sum_{n=0}^{\infty}\frac{\Jacobi{\alpha}{\beta}{n}@{x}}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+1}{n}}t^n
\frac{\pochhammer{\lambda+\frac{1}{2}}{n}}{\pochhammer{2\lambda}{n}}\Ultra{\lambda}{n}@{x}t^n
}</math></div>
}</math></div>
== Substitution(s) ==
<div align="left"><math>{\displaystyle R=\sqrt{1-2xt+t^2}}</math></div><br />


== Proof ==
== Proof ==
Line 17: Line 21:
== Symbols List ==
== Symbols List ==


<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/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#iii <math>{\displaystyle (a)_n}</math>]</span> : Pochhammer symbol : [http://dlmf.nist.gov/5.2#iii http://dlmf.nist.gov/5.2#iii]
<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 />


Line 33: Line 36:
<br /><div id="drmf_foot">
<br /><div id="drmf_foot">
<div id="alignleft"> << [[Formula:KLS:09.08:12|Formula:KLS:09.08:12]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:12|Formula:KLS:09.08:12]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:13|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:13|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:14|Formula:KLS:09.08:14]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:14|Formula:KLS:09.08:14]] >> </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 R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} \frac{\pochhammer{\lambda+\frac{1}{2}}{n}}{\pochhammer{2\lambda}{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}}}


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 \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.


Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Formula:KLS:09.08:13&oldid=1687"