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

In this paper we will discuss asymptotic properties of monic polynomials {S-n(lambda)(x)}n >= 0 orthogonal with respect to the Sobolev inner product < p, q >(S) = integral(R) p(x)q(x)d mu(0) + lambda integral(R) p'(x)q'(x)d mu(1), with lambda > 0, d mu(0) = e(-x2) dx, d(mu 1), d(mu 1) = x(2)+a/x(2)+b e(-x2) dx, a, b is an element of R+ and a not equal b. It is well known that (mu 0, mu 1) is a pair of symmetric (1, 1)-coherent measures. This means that there exist sequences {a(n)}(n is an element of N), {b(n)}(n is an element of N), a(n) not equal b(n) for every n is an element of N, such that the algebraic relation H-n(x) + b(n-2)H(n-2)(x) = Q(n)(x) + a(n-2)Q(n-2)(x), n >= 2, is satisfied, where {Q(n)(x)}(n >= 0) is the sequence of monic orthogonal polynomials associated with mu(1) and {H-n(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. (C) 2018 Elsevier Inc. All rights reserved.

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