On the Diophantine Equation Gn(x)=Gm(P(x)): Higher-Order Recurrences

Let K be a field of characteristic 0 and let (Gn(x))n=0 ∞be a linear recurring sequence of degree din K[x] defined by the initial terms G0,... ,Gd-1∈ K[x] and by the difference equation$G_{n+d}(x)=A_{d-1}(x)G_{n+d-1}(x)+\cdots +A_{0}(x)G_{n}(x),\quad \text{for}n\geq 0$, with A0,... ,Ad-1∈ K[x]. Finally, let P(x) be an element of K[x]. In this paper we are giving fairly general conditions depending only on G0,... ,Gd-1, on P, and on A0,... ,Ad-1under which the Diophantine equation Gn(x)=Gm(P(x)) has only finitely many solutions (n,m)∈ Z2,n,m≥ 0. Moreover, we are giving an upper bound for the number of solutions, which depends only on d. This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.