First Derivative Sequences of Extended Fibonacci and Lucas Polynomials

Abstract

In this article we conclude our investigation on the Fibonacci and Lucas derivative sequences by generalizing the sequences $\left\{ {F\begin{array}{*{20}{c}} {(1)} \\ n \\ \end{array} } \right\}$ and $\left\{ {L\begin{array}{*{20}{c}} {(1)} \\ n \\ \end{array} } \right\}$ and $\left\{ {G\begin{array}{*{20}{c}} {(1)} \\ n \\ \end{array} } \right\}$ and $\left\{ {H\begin{array}{*{20}{c}} {(1)} \\ n \\ \end{array} } \right\}$ studied in [1] (see also [2]) and [3], respectively. To do this, we consider the polynomials U _n (x;k,h,m) and V _n (x; k, h, m) (or simply U _n (x) and V _n (x), if no misunderstanding can arise) defined by the second-order recurrence relations

$$ {{U}_{{n + 2}}}(x) = {{x}^{k}}{{U}_{{n + 1}}}(x) + {{x}^{h}}{{U}_{n}}(x),[{{U}_{0}}(x) = 0;{{U}_{1}}(x) = {{x}^{m}}], $$

((1.1))

and

$$ {{V}_{{n + 2}}}(x) = {{x}^{k}}{{V}_{{n + 1}}}(x) + {{x}^{h}}{{V}_{n}}(x),[{{V}_{0}}(x) = 2;{{V}_{1}}(x) = {{x}^{m}}] $$

((1.2))

where x is a nonzero indeterminate, and k,h and m are integers. Observe that U _n (1;k,h,m) = U _n (x;0, 0, 0) = F _n and V _n (1;k,h,m) = V _n (x;0,0,0) = L _n , the n ^th Fibonacci and Lucas numbers, respectively. As an illustration, the polynomials U _n (x) and V _n (x) are shown below for the first few values of n.