On the Diophantine equation $G_n(x)=G_m(P(x))$: Higher-order recurrences

Abstract:

Let $\mathbf {K}$ be a field of characteristic $0$ and let $(G_{n}(x))_{n=0}^{\infty }$ be a linear recurring sequence of degree $d$ in $\mathbf {K}[x]$ defined by the initial terms $G_0,\ldots ,G_{d-1}\in \mathbf {K}[x]$ and by the difference equation \[ G_{n+d}(x)=A_{d-1}(x)G_{n+d-1}(x)+\cdots +A_0(x)G_{n}(x), \quad \mbox {for} n\geq 0,\] with $A_0,\ldots ,A_{d-1}\in \mathbf {K}[x]$. Finally, let $P(x)$ be an element of $\mathbf {K}[x]$. In this paper we are giving fairly general conditions depending only on $G_0,\ldots ,G_{d-1},$ on $P$, and on $A_0,\ldots ,A_{d-1}$ under which the Diophantine equation \[ G_{n}(x)=G_{m}(P(x))\] has only finitely many solutions $(n,m)\in \mathbb {Z}^{2},n,m\geq 0$. 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.