Abstract
We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recurrence relations, where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application, we study convergence of the ratios of families of functions satisfying finite recurrence relations with varying functional coefficients. For example, we explicitly describe the asymptotic ratio for two classes of biorthogonal polynomials introduced by Ismail and Masson.
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J. B. passed away unexpectedly on April 8, 2009 at the age of 40. We dedicate this paper (started jointly with J. B. in Spring 2004) to the memory of this talented and tragic human being. Rest in peace, Julius.
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Borcea, J., Friedland, S. & Shapiro, B. Parametric Poincaré-Perron theorem with applications. JAMA 113, 197–225 (2011). https://doi.org/10.1007/s11854-011-0004-0
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DOI: https://doi.org/10.1007/s11854-011-0004-0