Products of members of Lucas sequences with indices in an interval being a power

Let r and s be coprime nonzero integers with ∆ = r 2 + 4s = 0. Let α and β be the roots of the quadratic equation x 2 − rx − s = 0, and assume that α/β is not a root of 1. We make the convention that |α| ≥ |β|. Put (u n) n≥0 and (v n) n≥0 for the Lucas sequences of the first and second kind of roots α and β whose general terms are given by