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http://functions.wolfram.com/04.22.06.0003.01
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LucasL[\[Nu]] == Sum[(1/k!) (GoldenRatio^Subscript[\[Nu], 0] ArcCsch[2]^k +
(1/(GoldenRatio^Subscript[\[Nu], 0] 2))
(E^(Pi I Subscript[\[Nu], 0]) (Pi I - ArcCsch[2])^k +
((-1)^k (Pi I + ArcCsch[2])^k)/E^(Pi I Subscript[\[Nu], 0])))
(\[Nu] - Subscript[\[Nu], 0])^k, {k, 0, Infinity}]
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Cell[BoxData[RowBox[List[RowBox[List["LucasL", "[", "\[Nu]", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["GoldenRatio", SubscriptBox["\[Nu]", "0"]], " ", SuperscriptBox[RowBox[List["ArcCsch", "[", "2", "]"]], "k"]]], "+", RowBox[List[FractionBox[SuperscriptBox["GoldenRatio", RowBox[List["-", SubscriptBox["\[Nu]", "0"]]]], "2"], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", SubscriptBox["\[Nu]", "0"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "-", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]], "k"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", SubscriptBox["\[Nu]", "0"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]], "k"]]]]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]", "0"]]], ")"]], "k"]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox["L", LucasL] </annotation> </semantics> <mi> ν </mi> </msub> <mo>  </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <msup> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </msup> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", Function[List[], GoldenRatio]] </annotation> </semantics> <mrow> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mo> - </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <msub> <mi> ν </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LucasL </ci> <ci> ν </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> GoldenRatio </ci> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> <ci> k </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <ci> GoldenRatio </ci> <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> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <pi /> <imaginaryi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <imaginaryi /> <apply> <ci> Subscript </ci> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <pi /> <imaginaryi /> </apply> <apply> <arccsch /> <cn type='integer'> 2 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </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> <ci> k </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LucasL", "[", "\[Nu]_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["GoldenRatio", SubscriptBox["\[Nu]\[Nu]", "0"]], " ", SuperscriptBox[RowBox[List["ArcCsch", "[", "2", "]"]], "k"]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["GoldenRatio", RowBox[List["-", SubscriptBox["\[Nu]\[Nu]", "0"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "-", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]], "k"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", SubscriptBox["\[Nu]\[Nu]", "0"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["ArcCsch", "[", "2", "]"]]]], ")"]], "k"]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "-", SubscriptBox["\[Nu]\[Nu]", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
