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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/07.07.23.0003.01









  


  








Input Form





Sum[(1/n!^2) LegendreP[n, z] w^n, {n, 0, Infinity}] == Hypergeometric0F1[1, ((z - 1) w)/2] Hypergeometric0F1[1, ((z + 1) w)/2] /; -1 < z < 1 && Abs[w] < 1








Standard Form





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MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> w </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;0&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;1&quot;, Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;z&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot; 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&quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;z&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot; &quot;, &quot;w&quot;]], Hypergeometric0F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> z </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> w </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <power /> <ci> w </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Hypergeometric0F1 </ci> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> w </ci> </apply> </apply> <apply> <ci> Hypergeometric0F1 </ci> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <apply> <abs /> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29