UNIFORM ASYMPTOTIC APPROXIMATIONS FOR THE MEIXNER–SOBOLEV POLYNOMIALS

Abstract

We obtain uniform asymptotic approximations for the monic Meixner–Sobolev polynomials Sn(x). These approximations for n → ∞, are uniformly valid for x/n restricted to certain intervals, and are in terms of Airy functions. We also give asymptotic approximations for the location of the zeros of Sn(x), especially the small and the large zeros are discussed. As a limit case, we also give a new asymptotic approximation for the large zeros of the classical Meixner polynomials. The method is based on an integral representation in which a hypergeometric function appears in the integrand. After a transformation, the hypergeometric functions can be uniformly approximated by unity, and all that remains are simple integrals for which standard asymptotic methods are used. As far as we are aware, this is the first time that standard uniform asymptotic methods have been used for the Sobolev-class of orthogonal polynomials.