Abstract
Our goal is to find an asymptotic behavior as n→∞ of the orthogonal polynomials Pn(z) defined by Jacobi recurrence coefficients an (off-diagonal terms) and bn (diagonal terms). We consider the case an→∞, bn→∞ in such a way that ∑an−1<∞ (that is, the Carleman condition is violated) and γn:=2−1bn(anan−1)−1∕2→γ as n→∞. In the case |γ|≠1 asymptotic formulas for Pn(z) are known; they depend crucially on the sign of |γ|−1. We study the critical case |γ|=1. The formulas obtained are qualitatively different in the cases |γn|→1−0 and |γn|→1+0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of Pn(z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type.
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