Formula:KLS:09.08:13: Difference between revisions

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<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>
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<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 ==
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== 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 />


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<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>

Latest revision as of 08:35, 22 December 2019


R - 1 ⁢ ( 1 + R - x ⁢ t 2 ) 1 2 - λ = ∑ n = 0 ∞ ( λ + 1 2 ) n ( 2 ⁢ λ ) n ⁢ C n λ ⁡ ( x ) ⁢ t n superscript 𝑅 1 superscript 1 𝑅 𝑥 𝑡 2 1 2 𝜆 superscript subscript 𝑛 0 Pochhammer-symbol 𝜆 1 2 𝑛 Pochhammer-symbol 2 𝜆 𝑛 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle R^{-1}\left(\frac{1+R-xt}{2}\right)% ^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty}\frac{{\left(\lambda+\frac{1}{2}% \right)_{n}}}{{\left(2\lambda\right)_{n}}}C^{\lambda}_{n}\left(x\right)t^{n}}}}

Substitution(s)

R = 1 - 2 ⁢ x ⁢ t + t 2 𝑅 1 2 𝑥 𝑡 superscript 𝑡 2 {\displaystyle{\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

Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
C n μ subscript superscript 𝐶 𝜇 𝑛 {\displaystyle{\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

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