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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0028.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] == (((-1)^(\[Nu] + \[Lambda] - 1) 2^(1 - \[Nu] - 2 \[Lambda]) Sin[\[Nu] Pi] Gamma[\[Nu] + 2 \[Lambda]])/(Pi Gamma[\[Lambda]] Gamma[1 + \[Nu] + \[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])) (PolyGamma[k + 1] - PolyGamma[1/2 - k - \[Nu] - \[Lambda]] - PolyGamma[k + \[Nu] + 2 \[Lambda]] + PolyGamma[ k + 2 \[Nu] + 2 \[Lambda] + 1]) (2/(1 - z))^k, {k, 0, Infinity}] + (((-1)^(\[Nu] + \[Lambda] - 1) 2^(1 - \[Nu] - 2 \[Lambda]) Sin[\[Nu] Pi] Gamma[\[Nu] + 2 \[Lambda]])/(Pi Gamma[\[Lambda]] Gamma[1 + \[Nu] + \[Lambda]])) (z - 1)^(-\[Nu] - 2 \[Lambda]) Log[(z - 1)/2] Hypergeometric2F1[\[Nu] + 2 \[Lambda], 1/2 + \[Nu] + \[Lambda], 1 + 2 \[Nu] + 2 \[Lambda], 2/(1 - z)] + ((2^(1 - \[Nu] - 2 \[Lambda]) Sqrt[Pi])/(Gamma[\[Nu] + 1] Gamma[\[Lambda]])) (z - 1)^\[Nu] Sum[(((2 \[Nu] + 2 \[Lambda] - k - 1)! Pochhammer[-\[Nu], k])/ (k! Gamma[1/2 - k + \[Nu] + \[Lambda]])) (2/(1 - z))^k, {k, 0, 2 \[Nu] + 2 \[Lambda] - 1}] /; Element[\[Lambda] + \[Nu], Integers] && \[Lambda] + \[Nu] >= 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[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "+", "\[Lambda]", "-", "1"]]], SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]], " "]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]", "+", "\[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"]], "\[Infinity]"], 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"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "k", "-", "\[Nu]", "-", "\[Lambda]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "+", "\[Lambda]", "-", "1"]]], SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]], " "]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]", "+", "\[Lambda]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]]], RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", 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[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", SqrtBox["\[Pi]"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "\[Lambda]"]], "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "\[Lambda]"]], "-", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "k", "+", "\[Nu]", "+", "\[Lambda]"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["\[Nu]", "\[Element]", "Integers"]], "]"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "+", "\[Lambda]", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[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"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", 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[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "k", "-", "\[Nu]", "-", "\[Lambda]"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "\[Lambda]"]], "+", "1"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", "k"]], "]"]]]]]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]", "+", "\[Lambda]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "+", "\[Lambda]", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["z", "-", "1"]], "2"], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", 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["\[Pi]", " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]", "+", "\[Lambda]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", SqrtBox["\[Pi]"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "\[Lambda]"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "\[Lambda]"]], "-", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", RowBox[List["1", "-", "z"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "k", "+", "\[Nu]", "+", "\[Lambda]"]], "]"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["\[Lambda]", "+", "\[Nu]"]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]]

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

 2001-10-29