Asymptotics of Sobolev orthogonal polynomials for Hermite (1,1)-coherent pairs

Abstract

In this paper we will discuss asymptotic properties of monic polynomials {Snλ(x)}n≥0 orthogonal with respect to the Sobolev inner product〈p,q〉S=∫Rp(x)q(x)dμ0+λ∫Rp′(x)q′(x)dμ1, with λ>0, dμ0=e−x2dx, dμ1=x2+ax2+be−x2dx, a,b∈R+ and a≠b. It is well known that (μ0,μ1) is a pair of symmetric (1,1)-coherent measures. This means that there exist sequences {an}n∈N, {bn}n∈N, an≠bn for every n∈N, such that the algebraic relationHn(x)+bn−2Hn−2(x)=Qn(x)+an−2Qn−2(x),n≥2, is satisfied, where {Qn(x)}n≥0 is the sequence of monic orthogonal polynomials associated with μ1 and {Hn(x)}n≥0 is the sequence of monic Hermite polynomials. We will study the relative asymptotics for Sobolev scaled polynomials and we will obtain Mehler–Heine type formulas, among others.