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variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > Representations through more general functions > Through Meijer G > Classical cases for the direct function itself





http://functions.wolfram.com/07.06.26.0011.01









  


  










Input Form





Fibonacci[\[Nu], 2 I (1 + 2 z)] == (((Sin[Pi \[Nu]] - 2 I Cos[Pi \[Nu]])/(4 Sqrt[Pi])) Sin[Pi \[Nu]] MeijerG[{{\[Nu], -\[Nu]}, {}}, {{0}, {-(1/2)}}, z])/E^((I Pi \[Nu])/2) + ((I/(8 Sqrt[Pi] Sqrt[-z])) Sin[2 Pi \[Nu]] MeijerG[{{1/2 + \[Nu], 1/2 - \[Nu]}, {}}, {{0}, {1/2}}, z])/ E^((I Pi \[Nu])/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]_", ",", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "z_"]]]], ")"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], ")"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]", ",", RowBox[List["-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List["-", FractionBox["1", "2"]]], "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List["4", " ", SqrtBox["\[Pi]"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], ")"]]]]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", "\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List["8", " ", SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]










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





2001-10-29