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 Fibonacci

 http://functions.wolfram.com/07.06.06.0019.01

 Input Form

 Fibonacci[\[Nu], z] == ((1/z) (z Sin[(Pi \[Nu])/2]^2 - Sqrt[z^2] Cos[(Pi \[Nu])/2]^2) HypergeometricPFQ[{1 + \[Nu]/2, (1 + \[Nu])/2}, {1 + \[Nu]}, -(4/z^2)])/ (z^2)^((1 + \[Nu])/2) + (1/z) (z^2)^((\[Nu] - 1)/2) (z Sin[(Pi \[Nu])/2]^2 + Sqrt[z^2] Cos[(Pi \[Nu])/2]^2) HypergeometricPFQ[{1 - \[Nu]/2, (1 - \[Nu])/2}, {1 - \[Nu]}, -(4/z^2)] /; !IntervalMemberQ[Interval[{-2, 0}], I z] && !IntervalMemberQ[Interval[{0, 2}], I z] && !Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "z"], SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List["-", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]], "-", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]]]], ")"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", "\[Nu]"]], "}"]], ",", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "z"], SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], FractionBox[RowBox[List["\[Nu]", "-", "1"]], "2"]], RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]], "+", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Nu]"]], "}"]], ",", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "2"]], ",", "0"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "2"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 F TagBox["F", Fibonacci] ν ( z ) 1 z ( z 2 ) - ν + 1 2 ( z sin 2 ( π ν 2 ) - z 2 cos 2 ( π ν 2 ) ) 2 F 1 ( 1 + ν 2 , 1 + ν 2 ; 1 + ν ; - 4 z 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["1", "+", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "+", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 1 z ( z 2 ) ν - 1 2 ( z sin 2 ( π ν 2 ) + z 2 cos 2 ( π ν 2 ) ) 2 F 1 ( 1 - ν 2 , 1 - ν 2 ; 1 - ν ; - 4 z 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["1", "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; z ( - 2 , 0 ) z ( 0 , 2 ) ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] F TagBox["F", Fibonacci] ν ( z ) 1 z ( z 2 ) - ν + 1 2 ( z sin 2 ( π ν 2 ) - z 2 cos 2 ( π ν 2 ) ) 2 F 1 ( 1 + ν 2 , 1 + ν 2 ; 1 + ν ; - 4 z 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["1", "+", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "+", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 1 z ( z 2 ) ν - 1 2 ( z sin 2 ( π ν 2 ) + z 2 cos 2 ( π ν 2 ) ) 2 F 1 ( 1 - ν 2 , 1 - ν 2 ; 1 - ν ; - 4 z 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["1", "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; z ( - 2 , 0 ) z ( 0 , 2 ) ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]], "-", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", "\[Nu]"]], "}"]], ",", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]]]], "]"]]]], "z"], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], FractionBox[RowBox[List["\[Nu]", "-", "1"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]], "+", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Nu]"]], "}"]], ",", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]]]], "]"]]]], "z"]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "2"]], ",", "0"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "2"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]]

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

 2001-10-29