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

publication date

  • November 2018

start page

  • 601

end page

  • 621

issue

  • 1

volume

  • 467

international standard serial number (ISSN)

  • 0022-247X

electronic international standard serial number (EISSN)

  • 1096-0813

abstract

  • 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.

keywords

  • Hermite (1,1)-coherent pairs; Sobolev polynomials; Asymptotic properties