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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/07.06.26.0004.01









  


  










Input Form





Fibonacci[\[Nu], z] == (1/2) (\[Nu] ((-z) Cos[(Pi \[Nu])/2]^3 + Sqrt[-z^2] Sin[(Pi \[Nu])/2]^3) HypergeometricPFQ[{1 - \[Nu]/2, 1 + \[Nu]/2}, {3/2}, 1 + z^2/4] + (Sin[Pi \[Nu]]/Sqrt[4 + z^2]) (z Cos[(Pi \[Nu])/2] + Sqrt[-z^2] Sin[(Pi \[Nu])/2]) HypergeometricPFQ[ {(1 + \[Nu])/2, (1 - \[Nu])/2}, {1/2}, 1 + z^2/4])










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[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "z"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "3"]]], "+", RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "3"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["1", "+", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["1", "+", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]]], ")"]]]]]]]]










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





2001-10-29





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