On the growth of linear recurrences in function fields

Let $(G_n{)}_{n=0}^{\mathrm{\infty }}$ be a non-degenerate linear recurrence sequence with power sum representation $ G_n = a_1(n) \alpha_1^n + \cdots + a_t(n) \alpha_t^n $. In this paper we will prove a function field analogue of the well known result that in the number field case, under some non-restrictive conditions, for $n$ large enough the inequality $ \vert G_n\vert \geq \left( \max_{j=1,\ldots,t} \vert \alpha_j\vert \right)^{n(1-\varepsilon)} $ holds true.