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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,3,z] > Representations through more general functions > Through Meijer G > Classical cases involving products of Legendre P





http://functions.wolfram.com/07.09.26.0035.01









  


  








Input Form





(1/Sqrt[1 + z]) LegendreP[\[Nu], \[Mu], 3, Sqrt[1 + z]] LegendreP[\[Nu], \[Mu] + 1, 3, Sqrt[1 + z]] == (1/(Sqrt[Pi] Gamma[1 - \[Mu] + \[Nu]] Gamma[-\[Mu] - \[Nu]])) MeijerG[{{0, \[Nu] + 1/2, -(1/2) - \[Nu]}, {}}, {{-\[Mu] - 1/2}, {-(1/2), \[Mu] + 1/2}}, z] /; !IntervalMemberQ[Interval[{-1, 0}], z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List["1", "+", "z"]]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", SqrtBox[RowBox[List["1", "+", "z"]]]]], "]"]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["\[Mu]", "+", "1"]], ",", "3", ",", SqrtBox[RowBox[List["1", "+", "z"]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SqrtBox["\[Pi]"], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["\[Mu]", "+", FractionBox["1", "2"]]]]], "}"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]








MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", SqrtBox[RowBox[List["1", "+", "z_"]]]]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", RowBox[List["\[Mu]_", "+", "1"]], ",", "3", ",", SqrtBox[RowBox[List["1", "+", "z_"]]]]], "]"]]]], SqrtBox[RowBox[List["1", "+", "z_"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["\[Mu]", "+", FractionBox["1", "2"]]]]], "}"]]]], "}"]], ",", "z"]], "]"]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]"]], "]"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]


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





2001-10-29