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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0035.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] == (-((2^(-\[Nu] - 2 \[Lambda]) Gamma[-\[Nu] - \[Lambda]] Gamma[\[Nu] + 2 \[Lambda]] Sin[\[Nu] Pi])/(Pi Gamma[\[Lambda]]))) (z - 1)^(-\[Nu] - 2 \[Lambda]) Sum[((Pochhammer[1/2 + \[Nu] + \[Lambda], k] Pochhammer[ \[Nu] + 2 \[Lambda], k])/(k! Pochhammer[1 + 2 \[Nu] + 2 \[Lambda], k])) (2/(1 - z))^k, {k, 0, -(1/2) - \[Nu] - \[Lambda]}] + (((-1)^(1/2 - \[Nu] - \[Lambda]) 2^(\[Nu] + 1) Pi)/ (Gamma[\[Nu] + 1] Gamma[1 - \[Nu] - \[Lambda]] Gamma[\[Lambda]])) (z - 1)^\[Nu] Hypergeometric2F1[-\[Nu], 1/2 - \[Nu] - \[Lambda], 1 - 2 \[Nu] - 2 \[Lambda], 2/(1 - z)] /; Element[-\[Lambda] - \[Nu] - 1/2, Integers] && -\[Lambda] - \[Nu] - 1/2 >= 0 && !Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]", "-", "\[Lambda]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]", "+", "\[Lambda]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Nu]", "-", "\[Lambda]"]]], " ", SuperscriptBox["2", RowBox[List["\[Nu]", "+", "1"]]], "\[Pi]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]", "-", "\[Lambda]"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]", "-", "\[Lambda]"]], ",", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]], ",", FractionBox["2", RowBox[List["1", "-", "z"]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "\[Lambda]"]], "-", "\[Nu]", "-", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "]"]]]]]]]]

 MathML Form

 C ν λ ( z ) ( - 1 ) 1 2 - λ - ν 2 ν + 1 π Γ ( ν + 1 ) Γ ( 1 - λ - ν ) Γ ( λ ) ( z - 1 ) ν 2 F 1 ( - ν , 1 2 - λ - ν ; 1 - 2 λ - 2 ν ; 2 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]", "-", "\[Nu]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[FractionBox["2", RowBox[List["1", "-", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - 2 - 2 λ - ν Γ ( - λ - ν ) Γ ( 2 λ + ν ) sin ( ν π ) π Γ ( λ ) ( z - 1 ) - 2 λ - ν k = 0 - λ - ν - 1 2 ( λ + ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( 2 λ + ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! ( 2 λ + 2 ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], "k"], Pochhammer] ( 2 1 - z ) k /; - λ - ν - 1 2 ν Condition Subscript C ν λ z -1 1 2 -1 λ -1 ν 2 ν 1 Gamma ν 1 Gamma 1 -1 λ -1 ν Gamma λ -1 z -1 ν Hypergeometric2F1 -1 ν 1 2 -1 λ -1 ν 1 -1 2 λ -1 2 ν 2 1 -1 z -1 -1 2 -2 λ -1 ν Gamma -1 λ -1 ν Gamma 2 λ ν ν Gamma λ -1 z -1 -2 λ -1 ν k 0 -1 λ -1 ν -1 1 2 Pochhammer λ ν 1 2 k Pochhammer 2 λ ν k k Pochhammer 2 λ 2 ν 1 k -1 2 1 -1 z -1 k -1 λ -1 ν -1 1 2 ν [/itex]

 Rule Form

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

 2001-10-29