LucasL[n + m] == (1/2) (5 Fibonacci[n] Fibonacci[m] + LucasL[n] LucasL[m]) /; Element[n, Integers] && Element[m, Integers]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LucasL", "[", RowBox[List["n", "+", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["5", RowBox[List["Fibonacci", "[", "n", "]"]], RowBox[List["Fibonacci", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["LucasL", "[", "n", "]"]], RowBox[List["LucasL", "[", "m", "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> m </mi> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> n </mi> </msub> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mi> m </mi> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> L </mi> <annotation encoding='Mathematica'> TagBox[&quot;L&quot;, LucasL] </annotation> </semantics> <mi> n </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LucasL </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <ci> Fibonacci </ci> <ci> m </ci> </apply> <apply> <ci> Fibonacci </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <apply> <ci> LucasL </ci> <ci> m </ci> </apply> <apply> <ci> LucasL </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LucasL", "[", RowBox[List["n_", "+", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["5", " ", RowBox[List["Fibonacci", "[", "n", "]"]], " ", RowBox[List["Fibonacci", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["LucasL", "[", "n", "]"]], " ", RowBox[List["LucasL", "[", "m", "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[Element]", "Integers"]]]]]]]]]]

2007-05-02