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 Fibonacci

 http://functions.wolfram.com/07.06.06.0018.01

 Input Form

 Fibonacci[\[Nu], z] == ((1/z) (z Sin[(Pi \[Nu])/2]^2 - Sqrt[z^2] Cos[(Pi \[Nu])/2]^2) Sum[((Pochhammer[1 + \[Nu]/2, k] Pochhammer[(1 + \[Nu])/2, k])/ (k! Pochhammer[1 + \[Nu], k])) (-(4/z^2))^k, {k, 0, Infinity}])/ (z^2)^((1 + \[Nu])/2) + (1/z) (z^2)^((\[Nu] - 1)/2) (z Sin[(Pi \[Nu])/2]^2 + Sqrt[z^2] Cos[(Pi \[Nu])/2]^2) Sum[((Pochhammer[1 - \[Nu]/2, k] Pochhammer[(1 - \[Nu])/2, k])/ (k! Pochhammer[1 - \[Nu], k])) (-(4/z^2))^k, {k, 0, Infinity}] /; Abs[z] > 2 && !Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "z"], SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List["-", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]], "-", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox["1", "z"], SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], FractionBox[RowBox[List["\[Nu]", "-", "1"]], "2"]], RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]], "+", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "2"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 F TagBox["F", Fibonacci] ν ( z ) 1 z ( z 2 ) ν - 1 2 ( z sin 2 ( π ν 2 ) + z 2 cos 2 ( π ν 2 ) ) k = 0 4 k ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "k"], Pochhammer] ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ")"]], "k"], Pochhammer] k ! ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( - 1 z 2 ) k + 1 z ( z 2 ) - ν + 1 2 ( z sin 2 ( π ν 2 ) - z 2 cos 2 ( π ν 2 ) ) k = 0 4 k ( ν 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], ")"]], "k"], Pochhammer] ( ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], ")"]], "k"], Pochhammer] k ! ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ( - 1 z 2 ) k /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" > 2 ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Fibonacci ν z 1 z -1 z 2 ν -1 2 -1 z ν 2 -1 2 z 2 1 2 ν 2 -1 2 k 0 4 k Pochhammer 1 -1 ν 2 -1 k Pochhammer 1 -1 ν 2 -1 k k Pochhammer 1 -1 ν k -1 -1 1 z 2 -1 k 1 z -1 z 2 -1 ν 1 2 -1 z ν 2 -1 2 -1 z 2 1 2 ν 2 -1 2 k 0 4 k Pochhammer ν 2 -1 1 k Pochhammer ν 1 2 -1 k k Pochhammer ν 1 k -1 -1 1 z 2 -1 k z 2 ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]], "-", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]]]]]]]]], "z"], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], FractionBox[RowBox[List["\[Nu]", "-", "1"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", SuperscriptBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]], "+", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SuperscriptBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], "2"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]]]]]]]]], "z"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "2"]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]]

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

 2001-10-29