Asymptotics of orthogonal polynomials with asymptotic Freud‐like weights

Abstract

We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann-Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given. K E Y W O R D S asymptotic approximation, asymptotic Freud-like weight, continuous dual Hahn polynomials, Riemann-Hilbert problem ∑ =0 ( ) ( 2 ) ( + ) − ( − + ) − !( − )! , Stud Appl Math. 2020;144:133-163. wileyonlinelibrary.com/journal/sapm

Hardy’s inequalities in finite dimensional Hilbert spaces

“…The lower bound obtained as a consequence of Theorem 1.3, which appear on the left-hand side of (1.12) is amazingly close to the asymptotic value for d n in (1.13), which is derived via [1, Theorem 1.1] and [10]. Indeed, it is not difficult to verify that for their difference one has…”

“…Wang [10] about the asymptotic behaviour of those zeros, to establish a sharp result concerning inequality…”

“…The precise uniform asymptotics for the continuous dual Hahn polynomials was obtained very recently in [10]. Set…”

“…where w(x) is defined in [10] and it is always positive. Therefore, we need to calculate the argument of A(ix) for small values of x when a…”