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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,mu,2,z] > Representations through more general functions > Through hypergeometric functions > Involving 2F~1





http://functions.wolfram.com/05.07.26.0025.01









  


  










Input Form





LegendreP[n, \[Mu], 2, z] == (2^\[Mu] Pi ((-(z/(Gamma[(1/2) (-\[Mu] - n)] Gamma[(1/2) (1 - \[Mu] + n)]))) Hypergeometric2F1Regularized[(1/2) (1 - \[Mu] - n), (1/2) (2 - \[Mu] + n), 3/2, z^2] + (1/(Gamma[(1/2) (1 - \[Mu] - n)] Gamma[(1/2) (2 - \[Mu] + n)])) Hypergeometric2F1Regularized[(1/2) (1 - \[Mu] + n), (-(1/2)) (\[Mu] + n), 1/2, z^2]))/(1 - z^2)^(\[Mu]/2)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", "\[Mu]"], " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["z", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "n"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "+", "n"]], ")"]]]], "]"]]]]]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "-", "n"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Mu]", "+", "n"]], ")"]]]], ",", FractionBox["3", "2"], ",", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "-", "n"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Mu]", "+", "n"]], ")"]]]], "]"]]]]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "n"]], ")"]]]], ",", FractionBox["1", "2"], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]]]]]]










MathML Form







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</ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> &#956; </ci> </apply> <pi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> &#956; </ci> <ci> n </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", "\[Mu]"], " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["\[Mu]", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["z", " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "-", "n"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Mu]", "+", "n"]], ")"]]]], ",", FractionBox["3", "2"], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "n"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "+", "n"]], ")"]]]], "]"]]]]]]], "+", FractionBox[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "n"]], ")"]]]], ",", FractionBox["1", "2"], ",", SuperscriptBox["z", "2"]]], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "-", "n"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Mu]", "+", "n"]], ")"]]]], "]"]]]]]]], ")"]]]]]]]]










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





2007-05-02





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