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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0044.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] == ((2^(1 - 2 \[Lambda]) Cos[Pi (\[Lambda] + \[Nu])] Sin[Pi \[Nu]])/ (Pi^(3/2) Gamma[\[Lambda]])) Sum[((-1)^k/(2^k k!)) (2 I (-1)^k E^(I Pi (1/2 - \[Lambda]) Floor[Arg[z - x]/(2 Pi)]) Pi Floor[Arg[z - x]/(2 Pi)] Gamma[k - \[Nu]] Gamma[k + 2 \[Lambda] + \[Nu]] Hypergeometric2F1Regularized[k - \[Nu], k + 2 \[Lambda] + \[Nu], 1/2 + k + \[Lambda], (1/2) (1 + x)] - Exp[Pi I (1 - 2 \[Lambda]) Floor[Arg[z - x]/(2 Pi)]] MeijerG[{{1 - k + \[Nu], 1 - k - 2 \[Lambda] - \[Nu]}, {}}, {{0, 1/2 - k - \[Lambda]}, {}}, (1 + x)/2]) (z - x)^k, {k, 0, Infinity}] /; Element[x, Reals] && x < -1

 Standard Form

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 MathML Form

 C ν λ ( z ) 2 1 - 2 λ cos ( π ( λ + ν ) ) sin ( π ν ) π 3 / 2 Γ ( λ ) k = 0 ( - 1 ) k 2 - k k ! ( 2 π ( - 1 ) k π ( 1 2 - λ ) arg ( z - x ) 2 π arg ( z - x ) 2 π Γ ( k - ν ) Γ ( k + 2 λ + ν ) 2 F ~ 1 ( k - ν , k + 2 λ + ν ; k + λ + 1 2 ; x + 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["x", "+", "1"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - π ( 1 - 2 λ ) arg ( z - x ) 2 π G 2 , 2 2 , 2 ( x + 1 2 - k + ν + 1 , - k - 2 λ - ν + 1 0 , - k - λ + 1 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[RowBox[List["x", "+", "1"]], "2"], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[RowBox[List["-", "k"]], "+", "\[Nu]", "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "k"]], "-", RowBox[List["2", " ", "\[Lambda]"]], "-", "\[Nu]", "+", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "k"]], "-", "\[Lambda]", "+", FractionBox["1", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) ( z - x ) k /; x TagBox["\[DoubleStruckCapitalR]", Function[Reals]] x < - 1 Condition Subscript C ν λ z 2 1 -1 2 λ λ ν ν 3 2 Gamma λ -1 k 0 -1 k 2 -1 k k -1 2 -1 k 1 2 -1 λ z -1 x 2 -1 z -1 x 2 -1 Gamma k -1 ν Gamma k 2 λ ν Hypergeometric2F1Regularized k -1 ν k 2 λ ν k λ 1 2 x 1 2 -1 -1 1 -1 2 λ z -1 x 2 -1 MeijerG -1 k ν 1 -1 k -1 2 λ -1 ν 1 0 -1 k -1 λ 1 2 x 1 2 -1 z -1 x k x x -1 [/itex]

 Rule Form

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

 2007-05-02