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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0072.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] == ((2^\[Nu] Gamma[\[Lambda] + \[Nu]])/(Gamma[\[Lambda]] Gamma[1 + \[Nu]])) z^\[Nu] Hypergeometric2F1[-(\[Nu]/2), (1 - \[Nu])/2, 1 - \[Lambda] - \[Nu], 1/z^2] - ((2^(-2 \[Lambda] - \[Nu]) Sin[Pi \[Nu]] Gamma[-\[Lambda] - \[Nu]] Gamma[2 \[Lambda] + \[Nu]])/(Pi Gamma[\[Lambda]])) z^(-\[Nu] - 2 \[Lambda]) Hypergeometric2F1[\[Lambda] + \[Nu]/2, \[Lambda] + (1 + \[Nu])/2, 1 + \[Lambda] + \[Nu], 1/z^2] /; !Element[\[Lambda] + \[Nu], Integers] && !IntervalMemberQ[Interval[{-1, 0}], z]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]], SuperscriptBox["z", "\[Nu]"], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List["1", "-", "\[Lambda]", "-", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]], " "]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", "\[Lambda]"]]]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Lambda]", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["\[Lambda]", "+", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], ",", RowBox[List["1", "+", "\[Lambda]", "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]]]

 MathML Form

 C ν λ ( z ) 2 ν Γ ( λ + ν ) z ν Γ ( λ ) Γ ( ν + 1 ) 2 F 1 ( - ν 2 , 1 - ν 2 ; - λ - ν + 1 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - 2 - 2 λ - ν sin ( π ν ) Γ ( - λ - ν ) Γ ( 2 λ + ν ) z - 2 λ - ν π Γ ( λ ) 2 F 1 ( λ + ν 2 , λ + ν + 1 2 ; λ + ν + 1 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Lambda]", "+", FractionBox["\[Nu]", "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Lambda]", "+", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; λ + ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] z ( - 1 , 0 ) C ν λ ( z ) 2 ν Γ ( λ + ν ) z ν Γ ( λ ) Γ ( ν + 1 ) 2 F 1 ( - ν 2 , 1 - ν 2 ; - λ - ν + 1 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - 2 - 2 λ - ν sin ( π ν ) Γ ( - λ - ν ) Γ ( 2 λ + ν ) z - 2 λ - ν π Γ ( λ ) 2 F 1 ( λ + ν 2 , λ + ν + 1 2 ; λ + ν + 1 ; 1 z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Lambda]", "+", FractionBox["\[Nu]", "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Lambda]", "+", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Lambda]", "+", "\[Nu]", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["1", SuperscriptBox["z", "2"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; λ + ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] z ( - 1 , 0 ) [/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[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List["1", "-", "\[Lambda]", "-", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[Lambda]"]], "-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Lambda]", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["\[Lambda]", "+", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], ",", RowBox[List["1", "+", "\[Lambda]", "+", "\[Nu]"]], ",", FractionBox["1", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "\[Element]", "Integers"]]]], "&&", 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)

 2007-05-02