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Rigorous WKB for finite-order linear recurrence relations with smooth coefficients. (English) Zbl 0861.39007
The \(\varepsilon \to 0\) behaviour of recurrence relations of the type \[ \sum^l_{j=0} a_j (k\varepsilon, \varepsilon) y_{k+j} = 0, \quad k\in \mathbb{Z} \] is studied. The \(a_j\) are \(C^\infty\) functions in each variable on \(I\times [0, \varepsilon_0]\) for a bounded interval \(I\) and \(\varepsilon_0 > 0\). Under certain regularity assumptions the asymptotic behavior of the solutions of such recurrences are found. In particular, it is shown that the formal perturbation-series solutions are asymptotic to true solutions of these recurrence. Some applications are also discussed.

MSC:
39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
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