A Mehler-Heine-type formula for Hermite-Sobolev orthogonal polynomials

We consider a Sobolev inner product such as (f,g)s = integral f(x)g(x) dmu(0)(x) + lambda integral f'(x)g'(x)dmu(1)(x), lambda > 0, with (mu(0),mu(1)) being a symmetrically coherent pair of measures with unbounded support. Denote by Q(n) the orthogonal polynomials with respect to (1) and they are so-called Hermite-Sobolev orthogonal polynomials. We give a Mehler-Heine-type formula for Q(n) when mu(1) is the measure corresponding to Hermite weight on R, that is, dmu(1) = e(-x2) dx and as a consequence an asymptotic property of both the zeros and critical points of Q(n) is obtained, illustrated by numerical examples. Some remarks and numerical experiments are carried out for dmu(0) = e(-x2) dx. An upper bound for \Qn\ on R is also provided in both cases. (C) 2002 Elsevier Science B.V. All rights reserved.