Asymptotic properties of a family of orthogonal polynomial sequences

摘要

It is known that the nth denominators Qn(a, z) of a real J-fraction of the form form an orthogonal polynomial sequence (OPS) with respect to some distribution function ψ(t) on . In this paper we prove the asymptotic formula where the convergence is uniform on compact subsets of ⨍⨍ and Jv(w) denotes the Bessel function of the first kind of order v. The given proof is based on a separate convergence result for continued fractions and explicit formulae derived for the polynomials Qn(a,z). Examples include which the distribution function ψ(t) is a simple step function with infinitely many jumps.