Asymptotic properties of a family of orthogonal polynomial sequences

Abstract

It is known that the nth denominators Q n ( 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 R . In this paper we prove the asymptotic formula ▪ where the convergence is uniform on compact subsets of 0 < ⨍z⨍ < ∞ and J v ( 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 Q n ( a, z). Examples include 0F 1( 3 2 ; (16z) −1) = 2√z sinh((2√ which the distribution function ψ( t) is a simple step function with infinitely many jumps.