On the diophantine equation G_n(x) = G_m(P(x))

Let K be a field of characteristic 0 and let p, q, G₀, G₁, P ∈ K[x], deg P ≥ 1. Further, let the sequence of polynomials (G_n(x))_n=0^∞ be defined by the second order linear recurring sequence G_n+2(x) = p(x)G_n+1(x) + q(x)G_n(x), for n ≥ 0. In this paper we give conditions under which the diophantine equation G_n(x)= G_m(P(x)) has at most exp(10¹⁸) many solutions (n, m) ∈ ℤ², n,m ≥ 0. The proof uses a very recent result on S-unit equations over fields of characteristic 0 due to Evertse, Schlickewei and Schmidt. Under the same conditions we present also bounds for the cardinality of the set {(m,n) ∈ ℕ | m ≠ n, ∃ c ∈ K\{0} such that G_n (x) = c G_m(P(x))}. In the last part we specialize our results to certain families of orthogonal polynomials.