Plancherel-Rotach Asymptotic Expansion for Some Polynomials from Indeterminate Moment Problems

We study the Plancherel-Rotach asymptotics of four families of orthogonal polynomials: the Chen-Ismail polynomials, the Berg-Letessier-Valent polynomials, and the Conrad-Flajolet polynomials I and II. All these polynomials arise in indeterminate moment problems, and three of them are birth and death process polynomials with cubic or quartic rates. We employ a difference equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to a conjecture about large degree behavior of polynomials orthogonal with respect to solutions of indeterminate moment problems