Lucas Polynomial Sequence

Author

Eric W. Weisstein
April 7, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/LucasPolynomialSequence.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LucasPolynomialSequence.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Definition

solns=Solve[{a+b==p,a b==-q},{a,b}]

Simplify[a-b/.solns[[2]]]

Simplify[a b/.solns[[2]]]

Simplify[{LucasW[{p,q},n,x],Lucasw[{p,q},n,x]}]

Table

TableForm[Flatten/@FullSimplify[{#,LucasW[#,n,x],Lucasw[#,n,x]}&/@
{{x,1},{2x,1},{1,2x},{3,x-2},{2x,-1},{x+1,-x}},
x>0&&n∈Integers]]//TraditionalForm

Table[{Fibonacci[n,x],(((2^(-n)) ((((x+(Sqrt[(x^2)+4])))^n)-(((x-(Sqrt[(x^2)+4])))^n)))/(Sqrt[(x^2)+4]))},{n,4}]//Simplify