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 $(\mathcal{U}n){n\geqslant 0}$ 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 $\Delta = r^2 +4 s >0$.