On the Diophantine equation 𝐺_{𝑛}(𝑥)=𝐺_{𝑚}(𝑃(𝑥)): Higher-order recurrences

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