Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Fibonacci






Mathematica Notation

Traditional Notation









Integer Functions > Fibonacci[nu] > Identities > Functional identities > Relations of special kind





http://functions.wolfram.com/04.11.17.0012.01









  


  










Input Form





1/(Fibonacci[n + a] Fibonacci[n + b] Fibonacci[n + c]) == (-1)^(n + a)/(Fibonacci[b - a] Fibonacci[c - b] Fibonacci[n + a]) + (-1)^(n + b)/(Fibonacci[c - b] Fibonacci[a - b] Fibonacci[n + b]) + (-1)^(n + c)/(Fibonacci[a - c] Fibonacci[b - c] Fibonacci[n + c]) /; Element[n, Integers] && n >= 0 && Element[a, Integers] && a > 0 && Element[b, Integers] && b > 0 && Element[c, Integers] && c > 0 && a != b && a != c && b != c










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "a"]], "]"]], RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "b"]], "]"]], RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "c"]], "]"]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "a"]]], RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["b", "-", "a"]], "]"]], RowBox[List["Fibonacci", "[", RowBox[List["c", "-", "b"]], "]"]], RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "a"]], "]"]]]]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "b"]]], RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["c", "-", "b"]], "]"]], RowBox[List["Fibonacci", "[", RowBox[List["a", "-", "b"]], "]"]], RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "b"]], "]"]]]]], "+", FractionBox[RowBox[List[" ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "c"]]]]], RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["a", "-", "c"]], "]"]], RowBox[List["Fibonacci", "[", RowBox[List["b", "-", "c"]], "]"]], RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "c"]], "]"]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["a", "\[Element]", "Integers"]], "\[And]", RowBox[List["a", ">", "0"]], "\[And]", RowBox[List["b", "\[Element]", "Integers"]], "\[And]", RowBox[List["b", ">", "0"]], "\[And]", RowBox[List["c", "\[Element]", "Integers"]], "\[And]", RowBox[List["c", ">", "0"]], "\[And]", RowBox[List["a", "\[NotEqual]", "b"]], "\[And]", RowBox[List["a", "\[NotEqual]", "c"]], "\[And]", RowBox[List["b", "\[NotEqual]", "c"]]]]]]]]










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> <mn> 1 </mn> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> b </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> c </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> b </mi> <mo> + </mo> <mi> n </mi> </mrow> </msup> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> c </mi> <mo> - </mo> <mi> b </mi> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> b </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> </mrow> </mfrac> <mo> + </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> c </mi> <mo> + </mo> <mi> n </mi> </mrow> </msup> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> c </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> </mrow> </mfrac> <mo> + </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> </msup> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> c </mi> <mo> - </mo> <mi> b </mi> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> c </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#8800; </mo> <mi> b </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#8800; </mo> <mi> c </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8800; </mo> <mi> c </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> </apply> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> b </ci> <ci> n </ci> </apply> </apply> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> c </ci> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> b </ci> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> b </ci> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> c </ci> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> c </ci> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <ci> Fibonacci </ci> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> a </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> b </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> c </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <neq /> <ci> a </ci> <ci> b </ci> </apply> <apply> <neq /> <ci> a </ci> <ci> c </ci> </apply> <apply> <neq /> <ci> b </ci> <ci> c </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox["1", RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["n_", "+", "a_"]], "]"]], " ", RowBox[List["Fibonacci", "[", RowBox[List["n_", "+", "b_"]], "]"]], " ", RowBox[List["Fibonacci", "[", RowBox[List["n_", "+", "c_"]], "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "a"]]], RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["b", "-", "a"]], "]"]], " ", RowBox[List["Fibonacci", "[", RowBox[List["c", "-", "b"]], "]"]], " ", RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "a"]], "]"]]]]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "b"]]], RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["c", "-", "b"]], "]"]], " ", RowBox[List["Fibonacci", "[", RowBox[List["a", "-", "b"]], "]"]], " ", RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "b"]], "]"]]]]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "c"]]], RowBox[List[RowBox[List["Fibonacci", "[", RowBox[List["a", "-", "c"]], "]"]], " ", RowBox[List["Fibonacci", "[", RowBox[List["b", "-", "c"]], "]"]], " ", RowBox[List["Fibonacci", "[", RowBox[List["n", "+", "c"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["a", "\[Element]", "Integers"]], "&&", RowBox[List["a", ">", "0"]], "&&", RowBox[List["b", "\[Element]", "Integers"]], "&&", RowBox[List["b", ">", "0"]], "&&", RowBox[List["c", "\[Element]", "Integers"]], "&&", RowBox[List["c", ">", "0"]], "&&", RowBox[List["a", "\[NotEqual]", "b"]], "&&", RowBox[List["a", "\[NotEqual]", "c"]], "&&", RowBox[List["b", "\[NotEqual]", "c"]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.