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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,z] > Summation > Infinite summation





http://functions.wolfram.com/07.07.23.0005.01









  


  








Input Form





Sum[((Pochhammer[\[Gamma], n] Pochhammer[1 - \[Gamma], n])/n!^2) LegendreP[n, z] w^n, {n, 0, Infinity}] == Hypergeometric2F1[\[Gamma], 1 - \[Gamma], 1, (1 - Sqrt[w^2 - 2 w z + 1] - w)/2] Hypergeometric2F1[\[Gamma], 1 - \[Gamma], 1, (1 - Sqrt[w^2 - 2 w z + 1] + w)/2] /; -1 < z < 1 && Abs[w] < 1








Standard Form





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MathML Form







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</mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> w </mi> <mi> n </mi> </msup> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#947; </mi> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#947; </mi> </mrow> </mrow> <mo> ; </mo> <mn> 1 </mn> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mrow> <msup> <mi> w </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;\[Gamma]&quot;, Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Gamma]&quot;]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;1&quot;, Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; 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Rule Form





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Date Added to functions.wolfram.com (modification date)





2001-10-29