Lucas Polynomial Sequence
A Lucas polynomial sequence is a pair of generalized polynomials which generalize the Lucas sequence to polynomials is given by
 |  | ![(Delta^k(x)[a^n(x)-(-1)^kb^n(x)])/(Delta(x))](/images/equations/LucasPolynomialSequence/Inline3.gif) |
(1)
|
 |  | ![Delta^k(x)[a^n(x)+(-1)^kb^n(x)],](/images/equations/LucasPolynomialSequence/Inline6.gif) |
(2)
|
where
 |  |  |
(3)
|
 |  |  |
(4)
|
Solving for 
and 
and taking
the solution for 
with the 
sign gives
 |
(5)
|
(Horadam 1996). Setting 
gives
 |  |  |
(6)
|
 |  | ![Delta^k(x)[1+(-1)^k],](/images/equations/LucasPolynomialSequence/Inline23.gif) |
(7)
|
giving
 |  |  |
(8)
|
 |  |  |
(9)
|
The sequences most commonly considered have 
, giving
![W_n(x)=W_n^0(x)=(a^n(x)-b^n(x))/(a(x)-b(x))
=([p(x)+sqrt(p^2(x)+4q(x))]^n-[p(x)-sqrt(p^2(x)+4q(x))]^n)/(2^nsqrt(p^2(x)+4q(x)))
w_n(x)=w_n^0(x)=a^n(x)+b^n(x)
=([p(x)+sqrt(p^2(x)+4q(x))]^n+[p(x)-sqrt(p^2(x)+4q(x))]^n)/(2^n).](/images/equations/LucasPolynomialSequence/NumberedEquation2.gif) |
(10)
|
The 
polynomials satisfy the recurrence
relation
 |
(11)
|
Special cases of the 
and 
polynomials
are given in the following table.
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