Asymptotics of Sobolev orthogonal polynomials for Hermite coherent pairs

Let Qn be the polynomials orthogonal with respect to the Sobolev inner product(f,g)S=∫ fg dμ0+∫ f′g′ dμ1,being (μ0,μ1) a coherent pair where one of the measures is the Hermite measure. The outer relative asymptotics for Qn with respect to Hermite polynomials are found. On the other hand, we consider the Sobolev scaled polynomials and we obtain the Plancherel–Rotach asymptotics for those as well as a consequence about their zeros.