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variants of this functions
GegenbauerC






Mathematica Notation

Traditional Notation









Hypergeometric Functions > GegenbauerC[nu,lambda,z] > Series representations > Generalized power series > Expansions at z==infinity > For the function itself > Expansions in 1/(1-z)





http://functions.wolfram.com/07.14.06.0026.01









  


  










Input Form





GegenbauerC[\[Nu], \[Lambda], z] == ((2^\[Nu] Gamma[\[Nu] + \[Lambda]])/(Gamma[\[Nu] + 1] Gamma[\[Lambda]])) (z - 1)^\[Nu] Hypergeometric2F1[-\[Nu], 1/2 - \[Nu] - \[Lambda], 1 - 2 \[Nu] - 2 \[Lambda], 2/(1 - z)] - ((2^(-\[Nu] - 2 \[Lambda]) Sin[\[Nu] Pi])/(Pi Gamma[\[Lambda]])) Gamma[-\[Nu] - \[Lambda]] Gamma[\[Nu] + 2 \[Lambda]] (z - 1)^(-\[Nu] - 2 \[Lambda]) Hypergeometric2F1[\[Nu] + 2 \[Lambda], 1/2 + \[Nu] + \[Lambda], 1 + 2 \[Nu] + 2 \[Lambda], 2/(1 - z)] /; !IntervalMemberQ[Interval[{-1, 1}], z] && !Element[2 \[Lambda] + 2 \[Nu], Integers]










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "\[Lambda]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]", "-", "\[Lambda]"]], ",", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]", "+", "\[Lambda]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]], "&&", RowBox[List["!", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], "\[Element]", "Integers"]]]]]]]]]]]]










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





2001-10-29