On an asymptotic equality for reproducing kernels and sums of squares of orthonormal polynomials

Abstract

In a recent paper, the first author considered orthonormal polynomials {pn} associated with a symmetric measure with unbounded support and with recurrence relation xpn(x) = Anpn+1(x) + An-1pn-1(x), n ≥ 0. Under appropriate restrictions on {An}, the first author established the identity (Formula Presented) uniformly for x in compact subsets of the real line. Here, we establish and evaluate this limit for a class of even exponential weights, and also investigate analogues for weights on a finite interval, and for some non-even weights.