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variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Fibonacci[nu,z] > Differentiation > Symbolic differentiation > With respect to nu





http://functions.wolfram.com/07.06.20.0009.01









  


  










Input Form





D[Fibonacci[\[Nu], z], {\[Nu], n}] == ArcSinh[z/2]^n Fibonacci[\[Nu], z] - (2^(\[Nu] - 1)/((z + Sqrt[4 + z^2])^\[Nu] Sqrt[4 + z^2])) ((-1)^n E^(I Pi \[Nu]) ((-I) Pi + ArcSinh[z/2])^n + ((-1)^n (I Pi + ArcSinh[z/2])^n)/E^(I Pi \[Nu]) - ArcSinh[z/2]^n (E^(I Pi \[Nu]) + E^((-I) Pi \[Nu]))) /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> n </mi> </msup> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> &#957; </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <msup> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> <mtext> </mtext> </mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 4 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]_", ",", "n_"]], "}"]]]]], RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]], "n"], " ", RowBox[List["Fibonacci", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["-", "\[Nu]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "+", RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]]]], ")"]], "n"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]]]], ")"]], "n"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["ArcSinh", "[", FractionBox["z", "2"], "]"]], "n"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]]]]]], ")"]]]], SqrtBox[RowBox[List["4", "+", SuperscriptBox["z", "2"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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