Rigorous WKB for finite order linear recurrence relations with smooth coefficients

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We study the ε→ 0 behavior of recurrence relations of the type ∑ⱼ₌₀ˡ aⱼ(kε,ε)yₖ₊ⱼ=0, k∈ (l fixed). The aⱼ are C∞ functions in each variable on I× [0,₀] for a bounded interval I and 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 {(Fₘ(kε,ε))}ₘ₌₁... ₗ where the functions Fₘ are C∞ in each variable on the same domain as the aⱼ, showing in particular that the formal perturbation-series solutions are asymptotic to true solutions of these recurrences. Some applications are also briefly discussed.