Smallest zeros of some types of orthogonal polynomials: asymptotics

We establish Mehler–Heine-type formulas for orthogonal polynomials related to rational modifications of Hermite weight on the real line and for Hermite–Sobolev orthogonal polynomials. These formulas give us the asymptotic behaviour of the smallest zeros of the corresponding orthogonal polynomials. Furthermore, we solve a conjecture posed in a previous paper about the asymptotics of the smallest zeros of the Hermite–Sobolev polynomials as well as an open problem concerning the asymptotics of these Sobolev orthogonal polynomials.