D[GegenbauerC[\[Nu], z], {\[Nu], m}] == \[Nu]^(-m - 1) (E^(I m Pi) Gamma[1 + m, (-I) \[Nu] ArcCos[z]] + Gamma[1 + m, I \[Nu] ArcCos[z]]/E^(I m Pi)) /; Element[m, Integers] && m >= 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> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> m </mi> </msup> <mrow> <msubsup> <mi> C </mi> <mi> &#957; </mi> <mrow> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> &#957; </mi> <mi> m </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <msup> <mi> &#957; </mi> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> &#957; </ci> <degree> <ci> m </ci> </degree> </bvar> <apply> <ci> GegenbauerC </ci> <ci> &#957; </ci> <cn type='integer'> 0 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <ci> m </ci> <pi /> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> &#957; </ci> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> m </ci> <pi /> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <imaginaryi /> <ci> &#957; </ci> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29