Sum[((Gamma[n - k + 1] Gamma[k + \[Lambda]]^2 (2 k + 2 \[Lambda] - 1) (-1)^k 4^k)/Gamma[k + n + 2 \[Lambda]]) (Subscript[z, 1]^2 - 1)^(k/2) (Subscript[z, 2]^2 - 1)^(k/2) GegenbauerC[n - k, k + \[Lambda], Subscript[z, 1]] GegenbauerC[n - k, k + \[Lambda], Subscript[z, 2]] GegenbauerC[k, \[Lambda] - 1/2, \[Alpha]], {k, 0, n}] == ((4^(1 - \[Lambda]) Sqrt[Pi] Gamma[\[Lambda]])/Gamma[\[Lambda] - 1/2]) GegenbauerC[n, \[Lambda], Subscript[z, 1] Subscript[z, 2] - Sqrt[Subscript[z, 1]^2 - 1] Sqrt[Subscript[z, 2]^2 - 1] \[Alpha]]

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</mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> C </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> &#955; </mi> </mrow> </msubsup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <mi> C </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> &#955; 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</mo> <msqrt> <mrow> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <ci> &#955; </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> k </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> k </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <ci> k </ci> <ci> &#955; </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <ci> k </ci> <ci> &#955; </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> k </ci> </apply> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <ci> &#945; </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <ci> &#955; </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> n </ci> </apply> <ci> &#955; </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> &#945; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29