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 GegenbauerC

 http://functions.wolfram.com/07.14.20.0001.01

 Input Form

 D[GegenbauerC[\[Nu], \[Lambda], z], \[Nu]] == (-((2^(1 - 2 \[Lambda]) Sqrt[Pi] Gamma[\[Nu] + 2 \[Lambda]])/ (Gamma[\[Nu] + 1] Gamma[\[Lambda]]))) Hypergeometric2F1Regularized[ -\[Nu], \[Nu] + 2 \[Lambda], 1/2 + \[Lambda], (1 - z)/2] (PolyGamma[\[Nu] + 1] - PolyGamma[\[Nu] + 2 \[Lambda]]) + (((z - 1) Gamma[\[Nu] + 2 \[Lambda]])/((1 + 2 \[Lambda]) Gamma[\[Nu] + 1] Gamma[2 \[Lambda]])) ((\[Nu] + 2 \[Lambda]) HypergeometricPFQ[ {{1 - \[Nu], 1 + \[Nu] + 2 \[Lambda]}, {1}, {1, -\[Nu]}}, {{2, 3/2 + \[Lambda]}, {}, {1 - \[Nu]}}, (1 - z)/2, (1 - z)/2] + \[Nu] HypergeometricPFQ[{{1 - \[Nu], 1 + \[Nu] + 2 \[Lambda]}, {1}, {1, \[Nu] + 2 \[Lambda]}}, {{2, 3/2 + \[Lambda]}, {}, {1 + \[Nu] + 2 \[Lambda]}}, (1 - z)/2, (1 - z)/2])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "\[Nu]"], RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", " ", "\[Lambda]", ",", " ", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]], " "]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Lambda]"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ")"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "\[Nu]"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List[FractionBox["3", "2"], "+", "\[Lambda]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Nu]"]], "}"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]]]], "+", RowBox[List["\[Nu]", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List[FractionBox["3", "2"], "+", "\[Lambda]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "+", "\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"], ",", FractionBox[RowBox[List["1", "-", "z"]], "2"]]], "]"]]]]]], ")"]]]]]]]]]]

 MathML Form

 C ν λ ( z ) ν ( z - 1 ) Γ ( 2 λ + ν ) ( 2 λ + 1 ) Γ ( ν + 1 ) Γ ( 2 λ ) ( ν F 2 0 1 2 1 2 ( 1 - ν , 2 λ + ν + 1 ; 1 ; 1 , 2 λ + ν ; 2 , λ + 3 2 ; ; 2 λ + ν + 1 ; 1 - z 2 , 1 - z 2 ) + ( 2 λ + ν ) F 2 0 1 2 1 2 ( 1 - ν , 2 λ + ν + 1 ; 1 ; 1 , - ν ; 2 , λ + 3 2 ; ; 1 - ν ; 1 - z 2 , 1 - z 2 ) ) - 2 1 - 2 λ π Γ ( 2 λ + ν ) Γ ( ν + 1 ) Γ ( λ ) ( ψ TagBox["\[Psi]", PolyGamma] ( ν + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( 2 λ + ν ) ) 2 F ~ 1 ( - ν , 2 λ + ν ; λ + 1 2 ; 1 - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Lambda]", "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["1", "-", "z"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] FormBox RowBox FractionBox RowBox RowBox SubsuperscriptBox C ν λ ( z ) RowBox ν RowBox RowBox FractionBox RowBox RowBox ( RowBox z - 1 ) RowBox Γ ( RowBox RowBox 2 λ + ν ) RowBox RowBox ( RowBox RowBox 2 λ + 1 ) RowBox Γ ( RowBox ν + 1 ) RowBox Γ ( RowBox 2 λ ) RowBox ( RowBox RowBox ν RowBox SubsuperscriptBox F RowBox 2 0 1 RowBox 2 1 2 [ RowBox GridBox ErrorBox RowBox RowBox 1 - ν , RowBox RowBox RowBox 2 λ + ν + 1 ; 1 ; 1 , RowBox RowBox RowBox 2 λ + ν ; RowBox 2 , RowBox RowBox RowBox λ + FractionBox 3 2 ; ; RowBox RowBox 2 λ + ν + 1 ; FractionBox RowBox 1 - z 2 , FractionBox RowBox 1 - z 2 ] + RowBox RowBox ( RowBox RowBox 2 λ + ν ) RowBox SubsuperscriptBox F RowBox 2 0 1 RowBox 2 1 2 ( RowBox RowBox GridBox RowBox RowBox 1 - ν , RowBox RowBox RowBox 2 λ + ν + 1 ; 1 ; 1 , RowBox RowBox - ν ; RowBox 2 , RowBox RowBox RowBox λ + FractionBox 3 2 ; ; RowBox 1 - ν ; FractionBox RowBox 1 - z 2 , FractionBox RowBox 1 - z 2 ) ) - RowBox FractionBox RowBox SuperscriptBox 2 RowBox 1 - RowBox 2 λ SqrtBox π RowBox Γ ( RowBox RowBox 2 λ + ν ) RowBox RowBox Γ ( RowBox ν + 1 ) RowBox Γ ( λ ) RowBox ( RowBox RowBox TagBox ψ PolyGamma ( RowBox ν + 1 ) - RowBox TagBox ψ PolyGamma ( RowBox RowBox 2 λ + ν ) ) TagBox TagBox RowBox RowBox SubscriptBox FormBox 2 TraditionalForm SubscriptBox OverscriptBox F ~ FormBox 1 TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox RowBox - ν Hypergeometric2F1Regularized Rule Editable , TagBox RowBox RowBox 2 λ + ν Hypergeometric2F1Regularized Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1Regularized Rule Editable ; TagBox TagBox TagBox RowBox λ + FractionBox 1 2 Hypergeometric2F1Regularized Rule Editable InterpretTemplate Function SlotSequence 1 Hypergeometric2F1Regularized Rule Editable ; TagBox FractionBox RowBox 1 - z 2 Hypergeometric2F1Regularized Rule Editable ) InterpretTemplate Function HypergeometricPFQRegularized Slot 1 Slot 2 Slot 3 Rule Editable Hypergeometric2F1Regularized TraditionalForm [/itex]

 Rule Form

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

 2001-10-29