GegenbauerC[\[Nu], \[Lambda], z] \[Proportional] Piecewise[{{0, Element[-\[Nu], Integers] && -\[Nu] > 0}, {ComplexInfinity, Element[-\[Nu] - 2 \[Lambda], Integers] && -\[Nu] - 2 \[Lambda] >= 0}, {-((2^(1/2 - \[Lambda]) Sin[Pi \[Nu]] Gamma[-(1/2) + \[Lambda]] (1 + z)^(1/2 - \[Lambda]))/(Sqrt[Pi] Gamma[\[Lambda]])) + ((Sin[\[Nu] Pi] Gamma[\[Nu] + 2 \[Lambda]])/(Pi Gamma[\[Nu] + 1] Gamma[2 \[Lambda]])) (EulerGamma + Log[(1 + z)/2] - PolyGamma[1/2 + \[Lambda]] + PolyGamma[-\[Nu]] + PolyGamma[2 \[Lambda] + \[Nu]]), Element[\[Lambda] - 3/2, Integers] && \[Lambda] - 3/2 >= 0 && !(Element[\[Nu], Integers] && \[Nu] >= 0)}, {(Sin[\[Nu] Pi]/Pi) (Log[(z + 1)/2] + 2 EulerGamma + PolyGamma[-\[Nu]] + PolyGamma[1 + \[Nu]]), \[Lambda] == 1/2 && !(Element[\[Nu], Integers] && \[Nu] >= 0)}, {(2^(1 - 2 \[Lambda]) Cos[Pi (\[Nu] + \[Lambda])] (-(1/2) - \[Lambda])! Gamma[\[Nu] + 2 \[Lambda]])/(Sqrt[Pi] Gamma[\[Lambda]] Gamma[\[Nu] + 1]) + (-((2^(1/2 - \[Lambda]) Cos[Pi (\[Nu] + \[Lambda])])/ ((1/2 - \[Lambda])! Sqrt[Pi] Gamma[\[Lambda]]))) (z + 1)^(1/2 - \[Lambda]) (EulerGamma + Log[(1 + z)/2] - PolyGamma[3/2 - \[Lambda]] + PolyGamma[1/2 - \[Lambda] - \[Nu]] + PolyGamma[1/2 + \[Lambda] + \[Nu]]), Element[-(1/2) - \[Lambda], Integers] && -(1/2) - \[Lambda] >= 0 && !(Element[\[Nu], Integers] && \[Nu] >= 0)}}, -((2^(1/2 - \[Lambda]) Sin[Pi \[Nu]] Gamma[-(1/2) + \[Lambda]] (1 + z)^(1/2 - \[Lambda]))/(Sqrt[Pi] Gamma[\[Lambda]])) + (2^(1 - 2 \[Lambda]) Cos[Pi (\[Lambda] + \[Nu])] Gamma[1/2 - \[Lambda]] Gamma[2 \[Lambda] + \[Nu]])/(Sqrt[Pi] Gamma[\[Lambda]] Gamma[1 + \[Nu]])] /; (z -> -1)

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RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "1"]]]], ")"]]]]]]]]

2007-05-02