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variants of this functions
JacobiP






Mathematica Notation

Traditional Notation









Polynomials > JacobiP[n,a,b,z] > Summation > Infinite summation





http://functions.wolfram.com/05.06.23.0003.01









  


  










Input Form





Sum[((Pochhammer[a, k] Pochhammer[1 + \[Alpha] + \[Beta] - a, k])/ (Pochhammer[1 + \[Alpha], k] Pochhammer[1 + \[Beta], k])) JacobiP[k, \[Alpha], \[Beta], z] t^k, {k, 0, Infinity}] == Hypergeometric2F1[a, 1 + \[Alpha] + \[Beta] - a, 1 + \[Alpha], (1/2) (1 - t - Sqrt[1 - 2 z t + t^2])] Hypergeometric2F1[a, 1 + \[Alpha] + \[Beta] - a, 1 + \[Beta], (1/2) (1 + t - Sqrt[1 - 2 z t + t^2])]










Standard Form





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MathML Form







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</mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> t </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </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;a&quot;, Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;-&quot;, &quot;a&quot;]], &quot;+&quot;, &quot;\[Alpha]&quot;, &quot;+&quot;, &quot;\[Beta]&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;\[Beta]&quot;, &quot;+&quot;, &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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</ci> <ci> &#946; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#946; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <ci> t </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a_", ",", "k_"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Alpha]_", "+", "\[Beta]_", "-", "a_"]], ",", "k_"]], "]"]]]], ")"]], " ", RowBox[List["JacobiP", "[", RowBox[List["k_", ",", "\[Alpha]_", ",", "\[Beta]_", ",", "z_"]], "]"]], " ", SuperscriptBox["t_", "k_"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Alpha]_"]], ",", "k_"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Beta]_"]], ",", "k_"]], "]"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", RowBox[List["1", "+", "\[Alpha]", "+", "\[Beta]", "-", "a"]], ",", RowBox[List["1", "+", "\[Alpha]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "t", "-", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", "z", " ", "t"]], "+", SuperscriptBox["t", "2"]]]]]], ")"]]]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", RowBox[List["1", "+", "\[Alpha]", "+", "\[Beta]", "-", "a"]], ",", RowBox[List["1", "+", "\[Beta]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "t", "-", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", "z", " ", "t"]], "+", SuperscriptBox["t", "2"]]]]]], ")"]]]]]], "]"]]]]]]]]










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





2002-12-18





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