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 GegenbauerC

 http://functions.wolfram.com/07.13.06.0048.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["2", "\[Pi]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "2"]], ")"]], "j"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["j", "2"], "-", FractionBox["\[Nu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["j", "2"], "+", FractionBox["\[Nu]", "2"]]], "]"]]]], RowBox[List[" ", RowBox[List["j", "!"]]]]], SuperscriptBox["z", "j"]]]]]]], "\[Equal]", RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "+", RowBox[List[FractionBox["1", RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", "m"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["1", "+", "m"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "m", "-", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "m", "+", "\[Nu]"]], ")"]]]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["m", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["m", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["m", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["m", "2"]]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["\[Pi]", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "2"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["m", "+", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "+", "m"]]], " ", SuperscriptBox["z", RowBox[List["m", "+", "2"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "m", "-", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "m", "+", "\[Nu]"]], ")"]]]], "]"]]]], " ", ")"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", FractionBox["m", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["m", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", FractionBox["m", "2"]]], ",", RowBox[List["2", "+", FractionBox["m", "2"]]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]]]], ")"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 C ν ( 0 ) ( z ) F ( z , ν ) /; ( ( F m ( z , ν ) - sin ( π ν ) 2 π j = 0 m ( - 2 ) j Γ ( j 2 - ν 2 ) Γ ( j 2 + ν 2 ) j ! z j C ν ( 0 ) ( z ) - 1 π ( m + 2 ) ! ( 2 m + 1 ( - 1 ) m + 1 z m + 2 sin ( π ν ) Γ ( 1 2 ( m - ν + 2 ) ) Γ ( 1 2 ( m + ν + 2 ) ) ) 3 F 2 ( 1 , m 2 - ν 2 + 1 , m 2 + ν 2 + 1 ; m 2 + 3 2 , m 2 + 2 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["m", "2"], "-", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["m", "2"], "+", FractionBox["\[Nu]", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["m", "2"], "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["m", "2"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SuperscriptBox["z", "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 1 π ( m + 1 ) ! ( 2 m ( - z ) m + 1 sin ( π ν ) Γ ( 1 2 ( m - ν + 1 ) ) Γ ( 1 2 ( m + ν + 1 ) ) ) 3 F 2 ( 1 , m 2 - ν 2 + 1 2 , m 2 + ν 2 + 1 2 ; m 2 + 1 , m 2 + 3 2 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["m", "2"], "-", FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["m", "2"], "+", FractionBox["\[Nu]", "2"], "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["m", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["m", "2"], "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SuperscriptBox["z", "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) m TagBox["\[DoubleStruckCapitalN]", Function[Integers]] ) Condition GegenbauerC ν 0 z Subscript F z ν Subscript F m z ν -1 ν 2 -1 j 0 m -2 j Gamma j 2 -1 -1 ν 2 -1 Gamma j 2 -1 ν 2 -1 j -1 z j GegenbauerC ν 0 z -1 1 m 2 -1 2 m 1 -1 m 1 z m 2 ν Gamma 1 2 m -1 ν 2 Gamma 1 2 m ν 2 HypergeometricPFQ 1 m 2 -1 -1 ν 2 -1 1 m 2 -1 ν 2 -1 1 m 2 -1 3 2 m 2 -1 2 z 2 1 m 1 -1 2 m -1 z m 1 ν Gamma 1 2 m -1 ν 1 Gamma 1 2 m ν 1 HypergeometricPFQ 1 m 2 -1 -1 ν 2 -1 1 2 m 2 -1 ν 2 -1 1 2 m 2 -1 1 m 2 -1 3 2 z 2 m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "m"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "2"]], ")"]], "j"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["j", "2"], "-", FractionBox["\[Nu]", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["j", "2"], "+", FractionBox["\[Nu]", "2"]]], "]"]]]], ")"]], " ", SuperscriptBox["z", "j"]]], RowBox[List["j", "!"]]]]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "\[Equal]", RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "m"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["1", "+", "m"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "m", "-", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "m", "+", "\[Nu]"]], ")"]]]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["m", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["m", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["m", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["m", "2"]]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["m", "+", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "+", "m"]]], " ", SuperscriptBox["z", RowBox[List["m", "+", "2"]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "m", "-", "\[Nu]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "m", "+", "\[Nu]"]], ")"]]]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", FractionBox["m", "2"], "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["m", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", FractionBox["m", "2"]]], ",", RowBox[List["2", "+", FractionBox["m", "2"]]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "2"]], ")"]], "!"]]]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02