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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0040.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] == ((2^(1 - 2 \[Lambda]) Sin[Pi \[Nu]])/(Sqrt[Pi] Gamma[\[Lambda]])) Sum[(1/(2^k k!)) (Gamma[k - \[Nu]] Gamma[k + 2 \[Lambda] + \[Nu]] Cos[Pi (\[Lambda] + \[Nu])] (2 I E^(I Pi (1/2 - \[Lambda]) Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) Floor[(Pi + Arg[1 + Subscript[z, 0]])/(2 Pi)] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)] - Sec[Pi \[Lambda]] (1/(1 + Subscript[z, 0]))^((1/2 - \[Lambda]) Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) (1 + Subscript[z, 0])^ ((1/2 - \[Lambda]) Floor[Arg[z - Subscript[z, 0]]/(2 Pi)])) Hypergeometric2F1Regularized[k - \[Nu], k + 2 \[Lambda] + \[Nu], 1/2 + k + \[Lambda], (1 + Subscript[z, 0])/2] + 2^(-(1/2) + k + \[Lambda]) Pi Sec[Pi \[Lambda]] (1/(1 + Subscript[z, 0]))^((1/2 - \[Lambda]) Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) (1 + Subscript[z, 0])^ (1/2 - k - \[Lambda] + (1/2 - \[Lambda]) Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) Hypergeometric2F1Regularized[1/2 + \[Lambda] + \[Nu], 1/2 - \[Lambda] - \[Nu], 3/2 - k - \[Lambda], (1 + Subscript[z, 0])/ 2]) (z - Subscript[z, 0])^k, {k, 0, Infinity}] /; !Element[1/2 - \[Lambda], Integers]

 Standard Form

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 MathML Form

 C ν λ ( z ) 2 1 - 2 λ sin ( π ν ) π Γ ( λ ) k = 0 2 - k k ! ( 2 k + λ - 1 2 π sec ( π λ ) ( z 0 + 1 ) 1 2 - k - λ + ( 1 2 - λ ) arg ( z - z 0 ) 2 π ( 1 z 0 + 1 ) ( 1 2 - λ ) arg ( z - z 0 ) 2 π 2 F ~ 1 ( λ + ν + 1 2 , - λ - ν + 1 2 ; - k - λ + 3 2 ; z 0 + 1 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["\[Lambda]", "+", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "k"]], "-", "\[Lambda]", "+", FractionBox["3", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] + cos ( π ( λ + ν ) ) Γ ( k - ν ) Γ ( k + 2 λ + ν ) ( 2 π ( 1 2 - λ ) arg ( z - z 0 ) 2 π arg ( z 0 + 1 ) + π 2 π arg ( z - z 0 ) 2 π - sec ( π λ ) ( 1 z 0 + 1 ) ( 1 2 - λ ) arg ( z - z 0 ) 2 π ( z 0 + 1 ) ( 1 2 - λ ) arg ( z - z 0 ) 2 π ) 2 F ~ 1 ( k - ν , k + 2 λ + ν ; k + λ + 1 2 ; z 0 + 1 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["k", "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", "\[Lambda]", "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] ) ( z - z 0 ) k /; 1 2 - λ TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Subscript C ν λ z 2 1 -1 2 λ ν 1 2 Gamma λ -1 k 0 2 -1 k k -1 2 k λ -1 1 2 λ Subscript z 0 1 1 2 -1 k -1 λ 1 2 -1 λ z -1 Subscript z 0 2 -1 1 Subscript z 0 1 -1 1 2 -1 λ z -1 Subscript z 0 2 -1 Hypergeometric2F1Regularized λ ν 1 2 -1 λ -1 ν 1 2 -1 k -1 λ 3 2 Subscript z 0 1 2 -1 λ ν Gamma k -1 ν Gamma k 2 λ ν 2 1 2 -1 λ z -1 Subscript z 0 2 -1 Subscript z 0 1 2 -1 z -1 Subscript z 0 2 -1 -1 λ 1 Subscript z 0 1 -1 1 2 -1 λ z -1 Subscript z 0 2 -1 Subscript z 0 1 1 2 -1 λ z -1 Subscript z 0 2 -1 Hypergeometric2F1Regularized k -1 ν k 2 λ ν k λ 1 2 Subscript z 0 1 2 -1 z -1 Subscript z 0 k 1 2 -1 λ [/itex]

 Rule Form

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

 2007-05-02