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variants of this functions
Fibonacci




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > Series representations > Generalized power series > Expansions at nu==0





http://functions.wolfram.com/07.06.06.0002.01









  


  








Input Form





Fibonacci[\[Nu], z] == (1/Sqrt[z^2 + 4]) Sum[(Log[w]^k/k!) (1 - ((-1)^k/2) ((1 - (I Pi)/Log[w])^k + (1 + (I Pi)/Log[w])^k)) \[Nu]^k, {k, 1, Infinity}] /; w == (z + Sqrt[z^2 + 4])/2








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[" ", SuperscriptBox[RowBox[List["Log", "[", "w", "]"]], "k"]]], RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List[" ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], "2"], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], RowBox[List["Log", "[", "w", "]"]]]]], ")"]], "k"], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], RowBox[List["Log", "[", "w", "]"]]]]], ")"]], "k"]]], ")"]]]]]], ")"]], SuperscriptBox["\[Nu]", "k"]]]]]]]]], "/;", RowBox[List["w", "\[Equal]", FractionBox[RowBox[List["z", "+", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]]], "2"]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <msup> <mi> log </mi> <mi> k </mi> </msup> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> w </mi> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Fibonacci </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ln /> <ci> w </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <apply> <ln /> <ci> w </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <apply> <ln /> <ci> w </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> &#957; </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <ci> w </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Log", "[", "w", "]"]], "k"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], RowBox[List["Log", "[", "w", "]"]]]]], ")"]], "k"], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], RowBox[List["Log", "[", "w", "]"]]]]], ")"]], "k"]]], ")"]]]]]], ")"]], " ", SuperscriptBox["\[Nu]", "k"]]], RowBox[List["k", "!"]]]]], SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]], "/;", RowBox[List["w", "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]]], ")"]]]]]]]]]]]]


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





2001-10-29