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

We consider a Sobolev inner product such as (1)(f,g)S=∫f(x)g(x) dμ0(x)+λ∫f′(x)g′(x) dμ1(x), λ>0,with (μ0,μ1) being a symmetrically coherent pair of measures with unbounded support. Denote by Qn the orthogonal polynomials with respect to (1) and they are so-called Hermite–Sobolev orthogonal polynomials. We give a Mehler–Heine-type formula for Qn when μ1 is the measure corresponding to Hermite weight on R, that is, dμ1=e−x2 dx and as a consequence an asymptotic property of both the zeros and critical points of Qn is obtained, illustrated by numerical examples. Some remarks and numerical experiments are carried out for dμ0=e−x2 dx. An upper bound for |Qn| on R is also provided in both cases.