Diophantine equations for second order recursive sequences of polynomials

Let B be a non‐zero integer. Define the sequence of polynomials nG(x) by G0(x) = 0, G1(x) = 1, Gn+1(x) = nxG(x) + BGn−1(x), n ∈ N. We prove that the diophantine equation mG(x) = nG(y) for m, n ≥ 3, m ≠ n, has only finitely many solutions.