Relationship between Vieta-Lucas polynomials and Lucas sequences

Abstract

Let $w_n=w_n(P,Q)$ be numerical sequences which satisfy the recursion relation \begin{equation*} w_{n+2}=Pw_{n+1}-Qw_n. \end{equation*} We consider two special cases $(w_0,w_1)=(0,1)$ and $(w_0,w_1)=(2,P)$ and we denote them by $U_n$ and $V_n$ respectively. Vieta-Lucas polynomial $V_n(X,1)$ is the polynomial of degree $n$. We show that the congruence equation $V_n(X,1)\equiv C \mod p$ has a solution if and only if $U_{(p-\epsilon)/d}(C+2,C+2)$ is divisible by $p$, where $\epsilon\in\{\pm 1\}$ depends on $C$ and $p$, and $d=\gcd(n,p-\epsilon)$.