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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 abounded interval $I$ and $\e_0>0$. Under certain regularity assumptions we findthe asymptotic behavior of the solutions of such recurrences. In typical casesthere exists a fundamental set of solutions in the form $\{\exp(\epiF_m(k\epsilon,\epsilon))\}_{m=1... l}$ where the functions $F_m$ are$C^{\infty}$ in each variable on the same domain as the $a_j$, showing inparticular that the formal perturbation-series solutions are asymptotic to truesolutions of these recurrences. Some applications are also briefly discussed.

Rigorous WKB for Finite-Order Linear Recurrence Relations with Smooth Coefficients