POLYNOMIAL-EXPONENTIAL EQUATIONS INVOLVING MULTI-RECURRENCES

In this paper we consider polynomial-exponential Diophantine equations of the form G(n)((0))y(d) + G(n)((1))y(d-1) + ... + G(n)((d-1))y+G(n)(d) = 0 where G(n)((i)) are multi-recurrences, i.e. polynomial-exponential functions in variables n = (n(1), ... , n(k)). Under suitable (but restrictive) conditions we prove that there are finitely many multi-recurrences H-n((1)), ... , H-n((s)) such that for all solutions (n(1), ... , n(k), y) is an element of N-k x Z we either have H-n((i)) = 0 y = H-n((j)) for certain 1 <= i, j <= 8, respectively. This generalizes earlier results of this type on such equations. The proof uses a recent result by Corvaja and Zannier.