Rigorous WKB for finite order linear recurrence relations with smooth coefficients

Abstract

We study the $\epsilon \to 0$ behavior of recurrence relations of the type $\sum_{j=0}^l a_j(k\epsilon,\epsilon)y_{k+j}=0,$ $k\in \zdd$ ($l$ fixed). The $a_j$ are $C^{\infty}$ functions in each variable on $I\times [0,\e_0]$ for a bounded interval $I$ and $\e_0>0$. Under certain regularity assumptions we find the asymptotic behavior of the solutions of such recurrences. In typical cases there exists a fundamental set of solutions in the form $\{\exp(\epi F_m(k\epsilon,\epsilon))\}_{m=1... l}$ where the functions $F_m$ are $C^{\infty}$ in each variable on the same domain as the $a_j$, showing in particular that the formal perturbation-series solutions are asymptotic to true solutions of these recurrences. Some applications are also briefly discussed.