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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0052.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "\[Nu]"]], ">", "0"]]]]]], "}"]], ",", "\[IndentingNewLine]", RowBox[List["{", RowBox[List["ComplexInfinity", " ", ",", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", "\[Lambda]"]]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", "\[Lambda]"]]]], "\[GreaterEqual]", "0"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Nu]", "-", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List["\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]], ")"]], "!"]]]]], ",", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Nu]", "2"], "+", "\[Lambda]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]

 MathML Form

 C ν λ ( z ) 0 - ν + ~ - 2 λ - ν ( - 1 ) ν 2 ( λ ) ν - ν 2 TagBox[SubscriptBox[RowBox[List["(", "\[Lambda]", ")"]], RowBox[List["\[Nu]", "-", RowBox[List["\[LeftFloor]", FractionBox["\[Nu]", "2"], "\[RightFloor]"]]]]], Pochhammer] ( 2 z ) ν - 2 ν 2 ν 2 ! ( ν - 2 ν 2 ) ! ν cos ( π ν 2 ) Γ ( λ + ν 2 ) Γ ( ν 2 + 1 ) Γ ( λ ) True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; ( z "\[Rule]" 0 ) Condition Proportional Subscript C ν λ z 0 -1 ν SuperPlus OverTilde -2 λ -1 ν -1 ν 2 -1 Pochhammer λ ν -1 ν 2 -1 2 z ν -1 2 ν 2 -1 ν 2 -1 ν -1 2 ν 2 -1 -1 ν ν 2 -1 Gamma λ ν 2 -1 Gamma ν 2 -1 1 Gamma λ -1 Rule z 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List["0", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "\[Nu]"]], ">", "0"]]]]], List["ComplexInfinity", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]], "\[GreaterEqual]", "0"]]]]], List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Nu]", "-", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List["\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]], ")"]], "!"]]]]], RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]]], List[FractionBox[RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Nu]", "2"], "+", "\[Lambda]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], 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]", "0"]], ")"]]]]]]]]

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

 2007-05-02