Lucas Polynomial Sequence

A pair of generalized Polynomials which generalize the Lucas Sequence to Polynomials is given by

$$W_n^k(x) = {\Delta^k(x)[a^n(x)-(-1)^kb^n(x)]\over\Delta(x)}$$ (1)

$$w_n^k(x) = \Delta^k(x)[a^n(x)+(-1)^kb^n(x)],$$ (2)

where

$$a(x)+b(x)=p(x)$$ (3)

$$a(x)b(x)=-q(x)$$ (4)

$$a(x)-b(x)=\sqrt{p^2(x)+4q(x)}\equiv\Delta(x)$$ (5)

(Horadam 1996). Setting $n=0$ gives

$$W_0^k(x) = \Delta^k(x){1-(-1)^k\over\Delta(x)}$$ (6)

$$w_0^k(x) = \Delta^k(x)[1+(-1)^k],$$ (7)

giving

$$W_0^0(x) = 0$$ (8)

$$w_0^0(x) = 2.$$ (9)

The sequences most commonly considered have $k=0$, giving

$$W_n(x) \equiv W_n^0(x)={a^n(x)-b^n(x)\over a(x)-b(x)}$$ (10)

$$w_n(x) \equiv w_n^0(x)=a^n(x)+b^n(x).$$ (11)

Special cases are given in the following table.

$p(x)$	$q(x)$	Polynomial 1	Polynomial 2
$x$	1	Fibonacci $F_n(x)$	Lucas $L_n(x)$
$2x$	1	Pell $P_n(x)$	Pell-Lucas $Q_n(x)$
1	$2x$	Jacobsthal $J_n(x)$	Jacobsthal $j_n(x)$
$3x$	$-2$	Fermat ${\mathcal F}_n(x)$	Fermat-Lucas $f_n(x)$
$2x$	$-1$	Chebyshev Polynomial of the Second Kind $U_{n-1}(x)$	Chebyshev Polynomial of the First Kind $2T_n(x)$

See also Lucas Sequence

References

Horadam, A. F. ``Extension of a Synthesis for a Class of Polynomial Sequences.'' Fib. Quart. 34, 68-74, 1996.