z p[n, z] == p[n + 1, z] + ((n (2 \[Lambda] + n - 1))/ (4 (\[Lambda] + n - 1) (\[Lambda] + n))) p[n - 1, z] /; p[n, z] == (n!/(2^n Pochhammer[\[Lambda], n])) GegenbauerC[n, \[Lambda], z] && n > 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> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> p </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> &#955; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> p </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> p </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> p </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mrow> <msup> <mn> 2 </mn> <mi> n </mi> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> &#955; </mi> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;\[Lambda]&quot;, &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> C </mi> <mi> n </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <ci> z </ci> <apply> <ci> p </ci> <ci> n </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> n </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> n </ci> <ci> &#955; </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> n </ci> <ci> &#955; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> p </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <ci> p </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> p </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> &#955; </ci> <ci> n </ci> </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> <ci> z </ci> </apply> </apply> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29