Plancherel-Rotach type asymptotics for solutions of linear recurrence relations with rational coefficients

The asymptotic behaviour of solutions of difference equations with respect to the variable $n$ with spectral parameter $x$ is investigated. A new method for finding asymptotic expansions for basis solutions in overlapping domains of the $(n,x)$-space which extend to infinity is proposed. In principle, matching the expansions in the intersection of these domains makes it possible to determine the global asymptotic picture of the behaviour of solutions of equations in the complex plane of the spectral parameter $x$ for suitable scaling depending on $n$. The potential of the method is demonstrated using the examples of the Hermite and Meixner polynomials.
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