Asymptotic properties of Laguerre–Sobolev type orthogonal polynomials

In this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev‐type inner product ${⟨p,q⟩}_S={\int <!--\; ?\; -->}_0^{\mathrm{\infty <!--\; 8\; -->}}p(x)q(x)x^{\mathrm{\alpha <!--\; a\; -->}}{e}^{-<!--\; -\; -->x}dx+N{p}^{\mathrm{\prime <!--\; \text{'}\; -->}}(a)q^{\mathrm{\prime <!--\; \text{'}\; -->}}(a),\mathrm{\alpha <!--\; a\; -->}>-<!--\; -\; -->1$ where N ∈ ℝ + , and a ∈ ℝ − . We study the outer relative asymptotics of these polynomials with respect to the standard Laguerre polynomials. The analogue of the Mehler–Heine formula as well as a Plancherel–Rotach formula for the rescaled polynomials are given. The behavior of their zeros is also analyzed in terms of their dependence on N.