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 Fibonacci

 http://functions.wolfram.com/05.12.21.0003.01

 Input Form

 Integrate[z^(\[Alpha] - 1) Fibonacci[n, z], z] == -((2^(\[Alpha] - n - 1) z^\[Alpha] (z + Sqrt[4 + z^2])^(1 - n) (4^n (1 - \[Alpha] + n) Cos[Pi n] Hypergeometric2F1[ (1 - \[Alpha] - n)/2, 1 - \[Alpha], (3 - \[Alpha] - n)/2, (1/4) (z + Sqrt[4 + z^2])^2] + (z + Sqrt[4 + z^2])^(2 n) (\[Alpha] + n - 1) Hypergeometric2F1[(1 - \[Alpha] + n)/2, 1 - \[Alpha], (3 - \[Alpha] + n)/2, (1/4) (z + Sqrt[4 + z^2])^2]))/ ((-z) (z + Sqrt[4 + z^2]))^\[Alpha])/((1 - \[Alpha] + n) (\[Alpha] + n - 1))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["Fibonacci", "[", RowBox[List["n", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "n", "-", "1"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["1", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", "n"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "+", "n"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "n"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Alpha]", "-", "n"]], "2"], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", FractionBox[RowBox[List["3", "-", "\[Alpha]", "-", "n"]], "2"], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "n", "-", "1"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Alpha]", "+", "n"]], "2"], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", FractionBox[RowBox[List["3", "-", "\[Alpha]", "+", "n"]], "2"], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], "2"]]]]], "]"]]]]]], ")"]]]], ")"]]]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "n", "-", "1"]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z α - 1 F TagBox["F", Fibonacci] n ( z ) z - ( 2 α - n - 1 z α ( z + z 2 + 4 ) 1 - n ( - z ( z + z 2 + 4 ) ) - α ( ( α + n - 1 ) 2 F 1 ( 1 2 ( n - α + 1 ) , 1 - α ; 1 2 ( n - α + 3 ) ; 1 4 ( z + z 2 + 4 ) 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "-", "\[Alpha]", "+", "1"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Alpha]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "-", "\[Alpha]", "+", "3"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]]], ")"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ( z + z 2 + 4 ) 2 n + 4 n ( n - α + 1 ) cos ( π n ) 2 F 1 ( 1 2 ( 1 - α - n ) , 1 - α ; 1 2 ( 3 - α - n ) ; 1 4 ( z + z 2 + 4 ) 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "-", "n"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "\[Alpha]"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "\[Alpha]", "-", "n"]], ")"]]]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "+", "4"]]]]], ")"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] ) ) / ( ( - α + n + 1 ) ( α + n - 1 ) ) z z α -1 Fibonacci n z -1 2 α -1 n -1 z α z z 2 4 1 2 1 -1 n -1 z z z 2 4 1 2 -1 α α n -1 Hypergeometric2F1 1 2 n -1 α 1 1 -1 α 1 2 n -1 α 3 1 4 z z 2 4 1 2 2 z z 2 4 1 2 2 n 4 n n -1 α 1 n Hypergeometric2F1 1 2 1 -1 α -1 n 1 -1 α 1 2 3 -1 α -1 n 1 4 z z 2 4 1 2 2 -1 α n 1 α n -1 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Fibonacci", "[", RowBox[List["n_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "n", "-", "1"]]], " ", SuperscriptBox["z", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["1", "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["4", "n"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "+", "n"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "n"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "-", "n"]], ")"]]]], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "\[Alpha]", "-", "n"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "n", "-", "1"]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "+", "n"]], ")"]]]], ",", RowBox[List["1", "-", "\[Alpha]"]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "\[Alpha]", "+", "n"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], "2"]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Alpha]", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "n", "-", "1"]], ")"]]]]]]]]]]]

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

 2001-10-29