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variants of this functions
Fibonacci




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > Representations through more general functions > Through hypergeometric functions > Involving 2F1





http://functions.wolfram.com/07.06.26.0003.01









  


  








Input Form





Fibonacci[\[Nu], z] == (1/z) (((z Sin[(Pi \[Nu])/2]^2 - Sqrt[z^2] Cos[(Pi \[Nu])/2]^2) HypergeometricPFQ[{1 + \[Nu]/2, (1 + \[Nu])/2}, {1 + \[Nu]}, -(4/z^2)])/ (z^2)^((1 + \[Nu])/2)) + (1/z) ((z^2)^((\[Nu] - 1)/2) (z Sin[(Pi \[Nu])/2]^2 + Sqrt[z^2] Cos[(Pi \[Nu])/2]^2) HypergeometricPFQ[{1 - \[Nu]/2, (1 - \[Nu])/2}, {1 - \[Nu]}, -(4/z^2)]) /; !Element[\[Nu], Integers]








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]], "-", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", "\[Nu]"]], "}"]], ",", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]]]], "]"]]]], "z"], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], FractionBox[RowBox[List["\[Nu]", "-", "1"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]], "+", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Nu]"]], "}"]], ",", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]]]], "]"]]]], "z"]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]


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





2001-10-29