Formula:KLS:09.08:11: Difference between revisions

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<div id="alignleft"> << [[Formula:KLS:09.08:10|Formula:KLS:09.08:10]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:10|Formula:KLS:09.08:10]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:11|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:11|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:12|Formula:KLS:09.08:12]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:12|Formula:KLS:09.08:12]] >> </div>
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<br /><div align="center"><math>{\displaystyle  
<br /><div align="center"><math>{\displaystyle  
(1-x)^{\alpha}(1+x)^{\beta}\Jacobi{\alpha}{\beta}{n}@{x}=
(1-x^2)^{\lambda-\frac{1}{2}}\Ultra{\lambda}{n}@{x}=
\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n
\frac{\pochhammer{2\lambda}{n}(-1)^n}{\pochhammer{\lambda+\frac{1}{2}}{n}2^nn!}\left(\frac{d}{dx}\right)^n
\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]
\left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right]
}</math></div>
}</math></div>


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== Symbols List ==
== Symbols List ==


<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/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/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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<div id="alignleft"> << [[Formula:KLS:09.08:10|Formula:KLS:09.08:10]] </div>
<div id="alignleft"> << [[Formula:KLS:09.08:10|Formula:KLS:09.08:10]] </div>
<div id="aligncenter"> [[Jacobi#KLS:09.08:11|formula in Jacobi]] </div>
<div id="aligncenter"> [[Jacobi:_Special_cases#KLS:09.08:11|formula in Jacobi: Special cases]] </div>
<div id="alignright"> [[Formula:KLS:09.08:12|Formula:KLS:09.08:12]] >> </div>
<div id="alignright"> [[Formula:KLS:09.08:12|Formula:KLS:09.08:12]] >> </div>
</div>
</div>

Revision as of 00:34, 6 March 2017


( 1 - x 2 ) λ - 1 2 ⁢ C n λ ⁡ ( x ) = ( 2 ⁢ λ ) n ⁢ ( - 1 ) n ( λ + 1 2 ) n ⁢ 2 n ⁢ n ! ⁢ ( d d ⁢ x ) n ⁢ [ ( 1 - x 2 ) λ + n - 1 2 ] superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 Pochhammer-symbol 2 𝜆 𝑛 superscript 1 𝑛 Pochhammer-symbol 𝜆 1 2 𝑛 superscript 2 𝑛 𝑛 superscript 𝑑 𝑑 𝑥 𝑛 delimited-[] superscript 1 superscript 𝑥 2 𝜆 𝑛 1 2 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})^{\lambda-\frac{1}{2}}C^{% \lambda}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}(-1)^{n}}{{\left(% \lambda+\frac{1}{2}\right)_{n}}2^{n}n!}\left(\frac{d}{dx}\right)^{n}\left[(1-x% ^{2})^{\lambda+n-\frac{1}{2}}\right]}}}

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

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
( a ) n subscript 𝑎 𝑛 {\displaystyle{\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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