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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,z] > Generating functions





http://functions.wolfram.com/05.03.11.0002.01









  


  








Input Form





LegendreP[n, z] == SeriesTerm[(n!^2/(Pochhammer[a, n] Pochhammer[1 - a, n])) Hypergeometric2F1[a, 1 - a, 1, (1/2) (1 - t - Sqrt[1 - 2 z t + t^2])] Hypergeometric2F1[a, 1 - a, 1, (1/2) (1 + t - Sqrt[1 - 2 z t + t^2])], {t, 0, n}] /; -1 < z < -1








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["SeriesTerm", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["n", "!"]], ")"]], "2"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", RowBox[List["1", "-", "a"]], ",", "1", ",", 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", "-", "a"]], ",", "1", ",", 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[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "n"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "a"]], ",", "n"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["t", ",", "0", ",", "n"]], "}"]]]], "]"]], "/;", RowBox[List[RowBox[List["-", "1"]], "<", "z", "<", RowBox[List["-", "1"]]]]]]]]]]


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





2002-12-18