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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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