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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/07.06.26.0005.01









  


  








Input Form





Fibonacci[\[Nu], z] == (((I Sin[Pi \[Nu]])/(2 Sqrt[2 + I z])) Hypergeometric2F1[1/2 - \[Nu], 1/2 + \[Nu], 1/2, (2 + I z)/4])/ E^((I Pi \[Nu])/2) + ((\[Nu]/2) (Sin[Pi \[Nu]] - 2 I Cos[Pi \[Nu]]) Hypergeometric2F1[1 - \[Nu], 1 + \[Nu], 3/2, (2 + I z)/4])/ E^((I Pi \[Nu])/2)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[" ", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]], "2"]]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", FractionBox["1", "2"], ",", FractionBox[RowBox[List["2", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "4"]]], " ", "]"]]]], "+", RowBox[List[FractionBox["\[Nu]", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List[" ", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]], "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", FractionBox["3", "2"], ",", FractionBox[RowBox[List["2", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "4"]]], " ", "]"]]]]]]]]]]








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[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], ")"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], ")"]]]]]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Nu]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]], ",", FractionBox["3", "2"], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], ")"]]]]]], "]"]]]]]]]]]]


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





2001-10-29