On the Solutions of the Lucas Sequence Equation ±1Vn(P2,Q2)=∑∞k=1Uk−1(P1,Q1)xk

Suppose that {Un(P,Q)} and {Vn(P,Q)} are respectively the Lucas sequences of the first and second kinds with P≠0, Q≠0 and gcd(P,Q)=1. In this paper, we introduce an approach for studying the solutions (x,n) of the diophantine equation ±1Vn(P2,Q2)=∑k=1∞Uk−1(P1,Q1)xk, in the cases of (P1,Q1)≠(P2,Q2) and (P1,Q1)=(P2,Q2). Moreover, we apply the procedure of this approach with which −3≤P1,P2≤3, −2≤Q1≤2 and −1≤Q2≤1. Our approach is mainly based on transferring this equation into either an elliptic curve equation that can be solved easily using the Magma software, or a quadratic equation that can be solved using the quadratic formula.