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variants of this functions
Fibonacci




Mathematica Notation

Traditional Notation









Polynomials > Fibonacci[n,z] > Series representations > Generalized power series > Expansions at z==infinity





http://functions.wolfram.com/05.12.06.0014.01









  


  








Input Form





Fibonacci[n, z] \[Proportional] z^(-1 + n) (1 + (n - 2)/z^2 + ((-4 + n) (-3 + n))/(2 z^4) + \[Ellipsis]) /; (Abs[z] -> Infinity)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["n", "-", "2"]], SuperscriptBox["z", "2"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "n"]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[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> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Fibonacci </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -4 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["n", "-", "2"]], SuperscriptBox["z", "2"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "n"]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]


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





2001-10-29