Lucas Polynomial

Author

Eric W. Weisstein
October 27, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plot

V5

Plot[Evaluate[Table[LucasL[n,x],{n,0,5}]],{x,-1,1},
PlotStyle->{Black,Red,Yellow,Green,Blue,Violet},
AxesLabel->TraditionalForm/@{x,Subscript[L,n][x]}]

V7

Plot[Table[LucasL[n,x],{n,0,5}],{x,-2,2},
PlotStyle->{Black,Red,Yellow,Green,Blue,Purple},
AxesLabel->{x,LucasL[n,x]},Evaluated->True]

Enumeration

Table[LucasL[n,x],{n,0,5}]//ColumnForm//TraditionalForm

coef=CoefficientList[#,x]&/@(Table[LucasL[n,x],{n,0,5}]//Expand)

Flatten[coef]

Identities

(LucasL2[n, 1])==(LucasL[n])

LucasL[n,1]

(LucasL2[n, 0])==(((-1))^n)+1

FullSimplify[LucasL[n,0],n∈Integers]

Generating Function

GeneratingFunction[LucasL[n,x],n,t]

Sum[LucasL[n,x](t^n),{n,0,∞}]

Series[((1+(t^2))/(1-(t^2)-t x)),{t,0,10}]//Normal

Sum[LucasL[n,x](t^n),{n,0,10}]

SeriesCoefficient[((1+(t^2))/(1-(t^2)-t x)),{t,0,n}]//InputForm

FullSimplify[%]

(2^(-n)) ((((x-(Sqrt[4+(x^2)])))^n)+(((x+(Sqrt[4+(x^2)])))^n))//TraditionalForm

FullSimplify[(2^(-n)) ((((x-(Sqrt[4+(x^2)])))^n)+(((x+(Sqrt[4+(x^2)])))^n)),n∈Integers]

FullSimplify[(2^(-n)) ((((x-(Sqrt[4+(x^2)])))^n)+(((x+(Sqrt[4+(x^2)])))^n))-LucasL[n,x],n∈Integers]

Subtract@@@Table[{(2^(-n)) ((((x-(Sqrt[4+(x^2)])))^n)+(((x+(Sqrt[4+(x^2)])))^n)),LucasL[n,x]},{n,-10,10}]//FullSimplify

Table[{(2^(-n)) ((((x-(Sqrt[4+(x^2)])))^n)+(((x+(Sqrt[4+(x^2)])))^n)),LucasL[n,x]}/.{n->RandomComplex[],x->RandomComplex[]},{1}]

LucasL[.2,.2]

Integral and Derivative

Integral

Integrate[LucasL[n,x],x]

Integrate[FunctionExpand[LucasL[n,x]],x]

FunctionExpand[LucasL[n,x]]//FullSimplify

(((2^(-n)) (((x+(Sqrt[4+(x^2)])))^n) (-x+n (Sqrt[4+(x^2)])))/(-1+(n^2)))+(((2^n) (((x+(Sqrt[4+(x^2)])))^(-n)) (-x-n (Sqrt[4+(x^2)])) Cos[n π])/(-1+(n^2)))

Derivative

D[LucasL[n,x],x]

TraditionalForm[%]

Integrate[D[LucasL[n,x],x],x]