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 GegenbauerC

 http://functions.wolfram.com/07.14.06.0079.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2007-05-02