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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0058.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] \[Proportional] (-((Sqrt[Pi] (2 - 2 z)^(1/2 - \[Lambda]) Sec[Pi (\[Lambda] + \[Nu])] Sin[Pi \[Nu]])/(Gamma[\[Lambda]] Gamma[3/2 - \[Lambda]]))) (1 + (((-1 + 2 \[Lambda] + 2 \[Nu]) (1 + 2 \[Lambda] + 2 \[Nu]))/ (4 (3 - 2 \[Lambda]))) (z - 1) + (((-3 + 2 \[Lambda] + 2 \[Nu]) (-1 + 2 \[Lambda] + 2 \[Nu]) (1 + 2 \[Lambda] + 2 \[Nu]) (3 + 2 \[Lambda] + 2 \[Nu]))/ (32 (3 - 2 \[Lambda]) (5 - 2 \[Lambda]))) (z - 1)^2 + O[(z - 1)^3]) /; Element[-\[Lambda] - 1/2, Integers] && -\[Lambda] - 1/2 >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", "z"]]]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", "\[Lambda]"]], "]"]]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["4", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]]]]], RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["32", " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]], RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 C ν λ ( z ) - π ( 2 - 2 z ) 1 2 - λ sec ( π ( λ + ν ) ) sin ( π ν ) Γ ( λ ) Γ ( 3 2 - λ ) ( 1 + ( 2 λ + 2 ν - 1 ) ( 2 λ + 2 ν + 1 ) 4 ( 3 - 2 λ ) ( z - 1 ) + ( 2 λ + 2 ν - 3 ) ( 2 λ + 2 ν - 1 ) ( 2 λ + 2 ν + 1 ) ( 2 λ + 2 ν + 3 ) 32 ( 3 - 2 λ ) ( 5 - 2 λ ) ( z - 1 ) 2 + O ( ( z - 1 ) 3 ) ) /; - λ - 1 2 TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition Subscript C ν λ z -1 1 2 2 -1 2 z 1 2 -1 λ λ ν ν Gamma λ Gamma 3 2 -1 λ -1 1 2 λ 2 ν -1 2 λ 2 ν 1 4 3 -1 2 λ -1 z -1 2 λ 2 ν -3 2 λ 2 ν -1 2 λ 2 ν 1 2 λ 2 ν 3 32 3 -1 2 λ 5 -1 2 λ -1 z -1 2 O z -1 3 -1 λ -1 1 2 [/itex]

 Rule Form

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

 2007-05-02