Bartz-Marlewski equation with generalized Lucas components

Abstract

Let $\lbrace U_n\rbrace =\lbrace U_n(P,Q)\rbrace $ and $\lbrace V_n\rbrace =\lbrace V_n(P,Q)\rbrace $ be the Lucas sequences of the first and second kind respectively at the parameters $P \ge 1$ and $Q \in \lbrace -1, 1\rbrace $. In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation \[ x^2-3xy+y^2+x=0\,, \] where $(x,y)=(U_i, U_j)$ or $(V_i, V_j)$ with $i$, $ j \ge 1$. Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.