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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0065.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] \[Proportional] ((Cos[Pi (\[Nu] + \[Lambda])] Sec[Pi \[Lambda]] Gamma[\[Nu] + 2 \[Lambda]])/ (Gamma[\[Nu] + 1] Gamma[2 \[Lambda]])) (1 - ((\[Nu] (2 \[Lambda] + \[Nu]))/(2 \[Lambda] + 1)) (z + 1) - ((\[Nu] (1 - \[Nu]) (2 \[Lambda] + \[Nu]) (1 + 2 \[Lambda] + \[Nu]))/ (2 (2 \[Lambda] + 1) (2 \[Lambda] + 3))) (z + 1)^2 - O[(z + 1)^3]) - ((2^(1/2 - \[Lambda]) Sin[\[Nu] Pi] Gamma[\[Lambda] - 1/2])/ (Sqrt[Pi] Gamma[\[Lambda]])) (z + 1)^(1/2 - \[Lambda]) (1 + (((1 - 2 \[Lambda] - 2 \[Nu]) (1 + 2 \[Lambda] + 2 \[Nu]))/ (4 (3 - 2 \[Lambda]))) (z + 1) + (((1 - 2 \[Lambda] - 2 \[Nu]) (3 - 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]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Lambda]"]], ")"]]]], "]"]], " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Lambda]"]], "]"]]]]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]]]], RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]]], RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List["\[Nu]", RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", "3"]], ")"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], "-", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "3"], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", FractionBox["1", "2"]]], "]"]], " "]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[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["1", "-", RowBox[List["2", "\[Lambda]"]], "-", RowBox[List["2", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "-", 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["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["\[Lambda]", "+", FractionBox["1", "2"]]], ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

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

 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[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "\[Lambda]"]], ")"]]]], "]"]], " ", RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "3"]], ")"]]]]], "-", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "+", "1"]], "]"]], "3"]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Lambda]"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[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["(", RowBox[List["z", "+", "1"]], ")"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[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["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], RowBox[List["32", " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "+", "1"]], "]"]], "3"]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "/;", RowBox[List["!", RowBox[List[RowBox[List["\[Lambda]", "+", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02

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