Parrametric Poincare-Perron theorem with applications

TL;DR

This paper generalizes the Poincare-Perron theorem to recurrence relations with parameter-dependent coefficients and demonstrates applications to the asymptotic behavior of function ratios, including biorthogonal polynomials.

Contribution

It introduces a parametric version of the Poincare-Perron theorem and applies it to analyze convergence of ratios in families of functions with varying recurrence relations.

Findings

Established a uniform convergence criterion for recurrence coefficients.

Derived explicit asymptotic ratios for biorthogonal polynomial sequences.

Extended classical stability results to parameter-dependent recurrences.

Abstract

We prove a parametric generalization of the classical Poincare-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 sequences of biorthogonal polynomials introduced by Ismail and Masson.