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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 pFq





http://functions.wolfram.com/07.06.26.0007.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "-", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], "]"]]]], ")"]]]]]], "]"]]]], "-", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "-", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], "]"]]]], ")"]]]]]], "]"]], "+", RowBox[List["2", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], "]"]]]]]], "]"]]]]]], ")"]]]]]]]]








MathML Form







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</ci> <apply> <ln /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> HypergeometricPFQ </ci> <list /> <list /> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> HypergeometricPFQ </ci> <list /> <list /> <apply> <times /> <ci> &#957; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "-", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], "]"]]]], ")"]]]]]], "]"]]]], "-", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "-", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], "]"]]]], ")"]]]]]], "]"]], "+", RowBox[List["2", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], "]"]]]]]], "]"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]]]]]]]


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





2001-10-29