The Diophantine equation $a\left(\frac{b^k-1}{b-1}\right)={\mathcal{U}}_n-{\mathcal{U}}_m$

Abstract

Here, we find all positive integer solutions of the Diophantine equation in the title, where $({\mathcal{U}}_n{)}_{n⩾0}$ is the generalized Lucas sequence ${\mathcal{U}}_0=0,{\mathcal{U}}_1=1$ and ${\mathcal{U}}_{n+1}=r{\mathcal{U}}_n+s{\mathcal{U}}_{n-1}$ with $r$ and $s$ integers such that $\Delta =r^2+4s>0$.