Derivative Sequences of Fibonacci and Lucas Polynomials

Abstract

Let us consider the Fibonacci polynomials U _n(x) and the Lucas polynomials V _n (x) (or simply U _n and V_n, if there is no danger of confusion) defined as

$$ {U_n} = x{U_{n - 1}} + {U_{n - 2}}({U_0} = 0,{U_1} = 1) $$

(1.1)

and

$$ {V_n} = x{V_{n - 2}}({V_0} = 2,V = x) $$

(1.2)

where x is an indeterminate. These polynomials are a natural extension of the numbers U _n(m) and V _n(m) considered in [1]. They have already been considered elsewhere (e.g., see [6]).