Lucas numbers: Series representations (formula 04.22.06.0010)

LucasL

$Mathematica Notation$

Integer Functions

LucasL[nu]

Series representations

Asymptotic series expansions

http://functions.wolfram.com/04.22.06.0010.01

Input Form

LucasL[\[Nu]] \[Proportional] Piecewise[{{GoldenRatio^\[Nu], Im[\[Nu]] < 0 && Re[\[Nu]] - Pi Abs[Im[\[Nu]]] > 0}, {(1/2) E^(I \[Nu] Pi - \[Nu] ArcCsch[2]), Im[\[Nu]] < 0 && Re[\[Nu]] + Pi Im[\[Nu]] < 0}, {(1/2) E^((-I) Pi \[Nu] - \[Nu] ArcCsch[2]), Im[\[Nu]] > 0 && Re[\[Nu]] - Pi Im[\[Nu]] < 0}}, GoldenRatio^\[Nu] + Cos[\[Nu] Pi]/GoldenRatio^\[Nu]] /; (Abs[\[Nu]] -> Infinity)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LucasL", "[", "\[Nu]", "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SuperscriptBox["GoldenRatio", "\[Nu]"], ",", RowBox[List[RowBox[List[RowBox[List["Im", "[", "\[Nu]", "]"]], "<", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["Abs", "[", RowBox[List["Im", "[", "\[Nu]", "]"]], "]"]]]]]], ">", "0"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "-", RowBox[List["\[Nu]", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]]]]], ",", RowBox[List[RowBox[List[RowBox[List["Im", "[", "\[Nu]", "]"]], "<", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Im", "[", "\[Nu]", "]"]]]]]], "<", "0"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Nu]"]], "-", RowBox[List["\[Nu]", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]]]]], ",", RowBox[List[RowBox[List[RowBox[List["Im", "[", "\[Nu]", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["Im", "[", "\[Nu]", "]"]]]]]], "<", "0"]]]]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox["GoldenRatio", "\[Nu]"], " ", "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], SuperscriptBox["GoldenRatio", RowBox[List["-", "\[Nu]"]]]]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox["L", LucasL] </annotation> </semantics> <mi> ν </mi> </msub> <mo> ∝ </mo> <mrow> <mo>  </mo> <mtable> <mtr> <mtd> <msup> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <mi> ν </mi> </msup> </mtd> <mtd> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> - </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mrow> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <msup> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <mi> ν </mi> </msup> </mrow> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> ν </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> LucasL </ci> <ci> ν </ci> </apply> <piecewise> <piece> <apply> <power /> <ci> GoldenRatio </ci> <ci> ν </ci> </apply> <apply> <and /> <apply> <lt /> <apply> <imaginary /> <ci> ν </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <plus /> <apply> <real /> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <abs /> <apply> <imaginary /> <ci> ν </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </piece> <piece> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> ν </ci> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <imaginary /> <ci> ν </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <plus /> <apply> <real /> <ci> ν </ci> </apply> <apply> <times /> <pi /> <apply> <imaginary /> <ci> ν </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </piece> <piece> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> <ci> ν </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <imaginary /> <ci> ν </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <plus /> <apply> <real /> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <imaginary /> <ci> ν </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </piece> <otherwise> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> GoldenRatio </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <ci> ν </ci> <pi /> </apply> </apply> </apply> <apply> <power /> <ci> GoldenRatio </ci> <ci> ν </ci> </apply> </apply> </otherwise> </piecewise> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> ν </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LucasL", "[", "\[Nu]_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[SuperscriptBox["GoldenRatio", "\[Nu]"], RowBox[List[RowBox[List[RowBox[List["Im", "[", "\[Nu]", "]"]], "<", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["Abs", "[", RowBox[List["Im", "[", "\[Nu]", "]"]], "]"]]]]]], ">", "0"]]]]], List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Nu]", " ", "\[Pi]"]], "-", RowBox[List["\[Nu]", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["Im", "[", "\[Nu]", "]"]], "<", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Im", "[", "\[Nu]", "]"]]]]]], "<", "0"]]]]], List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Nu]"]], "-", RowBox[List["\[Nu]", " ", RowBox[List["ArcCsch", "[", "2", "]"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["Im", "[", "\[Nu]", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["Im", "[", "\[Nu]", "]"]]]]]], "<", "0"]]]]], List[RowBox[List[SuperscriptBox["GoldenRatio", "\[Nu]"], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox["GoldenRatio", RowBox[List["-", "\[Nu]"]]]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

2007-05-02