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 Vn, 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]).