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http://functions.wolfram.com/05.09.25.0006.01
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Integrate[(Sqrt[(m! (\[Lambda] + m) Gamma[\[Lambda]]^2)/
(Pi 2^(1 - 2 \[Lambda]) Gamma[2 \[Lambda] + m])]
(1 - t^2)^((2 \[Lambda] - 1)/4) GegenbauerC[m, \[Lambda], t])
(Sqrt[(n! (\[Lambda] + n) Gamma[\[Lambda]]^2)/(Pi 2^(1 - 2 \[Lambda])
Gamma[2 \[Lambda] + n])] (1 - t^2)^((2 \[Lambda] - 1)/4)
GegenbauerC[n, \[Lambda], t]), {t, -1, 1}] == KroneckerDelta[m, n] /;
Re[\[Lambda]] > -(1/2) && \[Lambda] != 0
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <mrow> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> C </mi> <mi> m </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> C </mi> <mi> n </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> λ </mi> <mo> ≠ </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> -1 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> m </ci> </apply> <apply> <plus /> <ci> m </ci> <ci> λ </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> λ </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> m </ci> </apply> <ci> λ </ci> </apply> <ci> t </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <plus /> <ci> n </ci> <ci> λ </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> λ </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> n </ci> </apply> <ci> λ </ci> </apply> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> λ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <neq /> <ci> λ </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
