Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Polynomials > Fibonacci[n,z] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself





http://functions.wolfram.com/05.12.13.0009.01









  


  










Input Form





Derivative[2][w][z] + (1/(4 + a^2 r^(2 z))) (-2 a^2 r^(2 z) (-Log[r] + Log[s]) - 4 (Log[r] + 2 Log[s])) Derivative[1][w][z] + (1/(4 + a^2 r^(2 z))) (4 Log[s] (Log[r] + Log[s]) - a^2 r^(2 z) ((-1 + n^2) Log[r]^2 + 2 Log[r] Log[s] - Log[s]^2)) w[z] == 0 /; w[z] == Subscript[c, 1] s^z Fibonacci[n, a r^z] + Subscript[c, 2] (s^z/(4 + a^2 r^(2 z))^(1/4)) LegendreP[-(1/2) + n, 1/2, 2, (I a r^z)/2]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "+", RowBox[List[FractionBox["1", RowBox[List["4", "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "r", "]"]]]], "+", RowBox[List["Log", "[", "s", "]"]]]], ")"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r", "]"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "s", "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["4", "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["Log", "[", "s", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r", "]"]], "+", RowBox[List["Log", "[", "s", "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["n", "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["Log", "[", "r", "]"]], "2"]]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "r", "]"]], " ", RowBox[List["Log", "[", "s", "]"]]]], "-", SuperscriptBox[RowBox[List["Log", "[", "s", "]"]], "2"]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", " ", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[SubscriptBox["c", "1"], SuperscriptBox["s", "z"], RowBox[List["Fibonacci", "[", RowBox[List["n", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", FractionBox[SuperscriptBox["s", "z"], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List[SuperscriptBox["a", "2"], SuperscriptBox["r", RowBox[List["2", "z"]]]]]]], ")"]], RowBox[List["1", "/", "4"]]]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], ",", FractionBox["1", "2"], ",", "2", ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", "a", " ", SuperscriptBox["r", "z"]]], "2"]]], "]"]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msup> <mi> w </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> s </mi> <mi> z </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> r </mi> <mi> z </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mi> s </mi> <mi> z </mi> </msup> <mtext> </mtext> </mrow> <mroot> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> r </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> <mn> 4 </mn> </mroot> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msubsup> <mo> ( </mo> <semantics> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> r </mi> <mi> z </mi> </msup> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; &quot;, &quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;a&quot;, &quot; &quot;, SuperscriptBox[&quot;r&quot;, &quot;z&quot;]]], HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <ln /> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> r </ci> </apply> </apply> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <ln /> <ci> r </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <ci> s </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ln /> <ci> s </ci> </apply> <apply> <plus /> <apply> <ln /> <ci> r </ci> </apply> <apply> <ln /> <ci> s </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> r </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <ci> s </ci> </apply> <apply> <ln /> <ci> r </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ln /> <ci> s </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> s </ci> <ci> z </ci> </apply> <apply> <ci> Fibonacci </ci> <ci> n </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> r </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <power /> <ci> s </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> r </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LegendreP </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <ci> a </ci> <apply> <power /> <ci> r </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "r_", "]"]]]], "+", RowBox[List["Log", "[", "s_", "]"]]]], ")"]]]], "-", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "s_", "]"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], RowBox[List["4", "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["Log", "[", "s_", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], "+", RowBox[List["Log", "[", "s_", "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["n_", "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["Log", "[", "r_", "]"]], "2"]]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "r_", "]"]], " ", RowBox[List["Log", "[", "s_", "]"]]]], "-", SuperscriptBox[RowBox[List["Log", "[", "s_", "]"]], "2"]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]], RowBox[List["4", "+", RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["r_", RowBox[List["2", " ", "z_"]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["s", "z"], " ", RowBox[List["Fibonacci", "[", RowBox[List["n", ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], "+", FractionBox[RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["s", "z"], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "n"]], ",", FractionBox["1", "2"], ",", "2", ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["4", "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["r", RowBox[List["2", " ", "z"]]]]]]], ")"]], RowBox[List["1", "/", "4"]]]]]]]]]]]]]]










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





2007-05-02