Fibonacci numbers: Series representations (formula 04.11.06.0016)

Fibonacci

$Mathematica Notation$

Integer Functions

Fibonacci[nu]

Series representations

Generalized power series

Expansions at generic point nu==nu₀

For the function itself

http://functions.wolfram.com/04.11.06.0016.01

Input Form

Fibonacci[\[Nu]] \[Proportional] Fibonacci[Subscript[\[Nu], 0]] + ((1/Sqrt[5]) ((1 + Sqrt[5])^(2 Subscript[\[Nu], 0]) ArcCsch[2] + 4^Subscript[\[Nu], 0] (ArcCsch[2] Cos[Pi Subscript[\[Nu], 0]] + Pi Sin[Pi Subscript[\[Nu], 0]])) (\[Nu] - Subscript[\[Nu], 0]))/ (2 (1 + Sqrt[5]))^Subscript[\[Nu], 0] + (1/10) (5 ArcCsch[2]^2 Fibonacci[Subscript[\[Nu], 0]] + Sqrt[5] ((1/2) (-1 + Sqrt[5]))^Subscript[\[Nu], 0] Pi (Pi Cos[Pi Subscript[\[Nu], 0]] - 2 ArcCsch[2] Sin[Pi Subscript[\[Nu], 0]])) (\[Nu] - Subscript[\[Nu], 0])^2 + O[(\[Nu] - Subscript[\[Nu], 0])^3]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Fibonacci", "[", "\[Nu]", "]"]], "\[Proportional]", RowBox[List[RowBox[List["Fibonacci", "[", SubscriptBox["\[Nu]", "0"], "]"]], "+", RowBox[List[FractionBox["1", SqrtBox["5"]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]]]], ")"]], RowBox[List["-", SubscriptBox["\[Nu]", "0"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List["2", " ", SubscriptBox["\[Nu]", "0"]]]], " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "+", RowBox[List[SuperscriptBox["4", SubscriptBox["\[Nu]", "0"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcCsch", "[", "2", "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "10"], " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", SuperscriptBox[RowBox[List["ArcCsch", "[", "2", "]"]], "2"], " ", RowBox[List["Fibonacci", "[", SubscriptBox["\[Nu]", "0"], "]"]]]], "+", RowBox[List[SqrtBox["5"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]]]], ")"]], SubscriptBox["\[Nu]", "0"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["ArcCsch", "[", "2", "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]], "2"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]], "3"], "]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <mi> ν </mi> </msub> <mo> ∝ </mo> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msub> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mn> 5 </mn> </msqrt> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mn> 4 </mn> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 10 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mn> 5 </mn> </msqrt> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msqrt> <mn> 5 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox["F", Fibonacci] </annotation> </semantics> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> Fibonacci </ci> <ci> ν </ci> </apply> <apply> <plus /> <apply> <ci> Fibonacci </ci> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> <apply> <cos /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <sin /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 10 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <apply> <plus /> <apply> <times /> <pi /> <apply> <cos /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> <apply> <sin /> <apply> <times /> <pi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Fibonacci </ci> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", "\[Nu]_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Fibonacci", "[", SubscriptBox["\[Nu]\[Nu]", "0"], "]"]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]]]], ")"]], RowBox[List["-", SubscriptBox["\[Nu]\[Nu]", "0"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], RowBox[List["2", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]]], " ", RowBox[List["ArcCsch", "[", "2", "]"]]]], "+", RowBox[List[SuperscriptBox["4", SubscriptBox["\[Nu]\[Nu]", "0"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["ArcCsch", "[", "2", "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]]]], SqrtBox["5"]], "+", RowBox[List[FractionBox["1", "10"], " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", SuperscriptBox[RowBox[List["ArcCsch", "[", "2", "]"]], "2"], " ", RowBox[List["Fibonacci", "[", SubscriptBox["\[Nu]\[Nu]", "0"], "]"]]]], "+", RowBox[List[SqrtBox["5"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]]]], ")"]], SubscriptBox["\[Nu]\[Nu]", "0"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["ArcCsch", "[", "2", "]"]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], "]"]], "3"]]]]]]]

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

2007-05-02

Fibonacci[n,z]
Fibonacci[nu,z]