Fibonacci function: Representations through more general functions (formula 07.06.26.0013)

Fibonacci

$Mathematica Notation$

Hypergeometric Functions

Fibonacci[nu,z]

Representations through more general functions

Through Meijer G

Classical cases for the direct function itself

http://functions.wolfram.com/07.06.26.0013.01

Input Form

Fibonacci[\[Nu], Sqrt[z]] == (Sin[Pi \[Nu]]/(2 Sqrt[Pi])) MeijerG[{{(1 + \[Nu])/2, (1 - \[Nu])/2}, {\[Nu]/2}}, {{0, 1/2}, {\[Nu]/2}}, z/4] /; !IntervalMemberQ[Interval[{-Infinity, 0}], z] && !Element[\[Nu], Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", SqrtBox["z"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", FractionBox["\[Nu]", "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["\[Nu]", "2"], "}"]]]], "}"]], ",", FractionBox["z", "4"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]]]

MathML Form

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Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", SqrtBox["z_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", FractionBox["\[Nu]", "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["\[Nu]", "2"], "}"]]]], "}"]], ",", FractionBox["z", "4"]]], "]"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]]

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

2001-10-29

Fibonacci[n,z]
Fibonacci[nu]