Asymptotic behavior of orthogonal polynomials without the Carleman condition

Abstract

Our goal is to find an asymptotic behavior as n→∞ of orthogonal polynomials Pn(z) defined by the Jacobi recurrence coefficients an,bn. We suppose that the off-diagonal coefficients an grow so rapidly that the series ∑an−1 converges, that is, the Carleman condition is violated. With respect to diagonal coefficients bn we assume that −bn(anan−1)−1/2→2β∞ for some β∞≠±1. The asymptotic formulas obtained for Pn(z) are quite different from the case ∑an−1=∞ when the Carleman condition is satisfied. In particular, if ∑an−1<∞, then the phase factors in these formulas do not depend on the spectral parameter z∈C. The asymptotic formulas obtained in the cases |β∞|<1 and |β∞|>1 are also qualitatively different from each other. As an application of these results, we find necessary and sufficient conditions for the essential self-adjointness of the corresponding minimal Jacobi operator.