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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0018.01

 Input Form

 GegenbauerC[\[Nu], \[Lambda], z] == ((Sin[\[Nu] Pi] Gamma[\[Nu] + 2 \[Lambda]])/(Pi Gamma[\[Nu] + 1] Gamma[2 \[Lambda]])) Log[(z + 1)/2] Hypergeometric2F1[-\[Nu], \[Nu] + 2 \[Lambda], 1/2 + \[Lambda], (z + 1)/2] - ((2^(1/2 - \[Lambda]) Sin[\[Nu] Pi] Gamma[\[Lambda] - 1/2])/ (Sqrt[Pi] Gamma[\[Lambda]])) (z + 1)^(1/2 - \[Lambda]) Sum[((Pochhammer[1/2 - \[Nu] - \[Lambda], k] Pochhammer[ 1/2 + \[Nu] + \[Lambda], k])/(k! Pochhammer[3/2 - \[Lambda], k])) ((z + 1)/2)^k, {k, 0, \[Lambda] - 3/2}] - ((2^(1 - 2 \[Lambda]) Sin[\[Nu] Pi] Gamma[\[Nu] + 2 \[Lambda]])/ (Sqrt[Pi] Gamma[\[Nu] + 1] Gamma[\[Lambda]])) Sum[((Pochhammer[-\[Nu], k] Pochhammer[\[Nu] + 2 \[Lambda], k])/ (k! Gamma[1/2 + k + \[Lambda]])) (PolyGamma[k + 1] - PolyGamma[k + \[Nu] + 2 \[Lambda]] - PolyGamma[k - \[Nu]] + PolyGamma[1/2 + k + \[Lambda]]) ((1 + z)/2)^k, {k, 0, Infinity}] /; Element[\[Lambda] - 3/2, Integers] && \[Lambda] - 3/2 >= 0 && !(Element[\[Nu], Integers] && \[Nu] >= 0)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", " ", "\[Lambda]"]], "]"]]]]], RowBox[List["Log", "[", FractionBox[RowBox[List["z", "+", "1"]], "2"], "]"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Lambda]"]], ",", FractionBox[RowBox[List["z", "+", "1"]], "2"]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "-", FractionBox["1", "2"]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], "-", "\[Lambda]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["\[Lambda]", "-", FractionBox["3", "2"]]]], 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[FractionBox["1", "2"], "+", "\[Nu]", "+", "\[Lambda]"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "-", "\[Lambda]"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "2"], ")"]], "k"]]]]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "\[Lambda]"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "\[Nu]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "+", "\[Lambda]"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "+", "z"]], "2"], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List["\[Lambda]", "-", FractionBox["3", "2"]]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["\[Lambda]", "-", FractionBox["3", "2"]]], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]], "]"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2001-10-29