Strong and Plancherel-Rotach asymptotics of non-diagonal Laguerre-Sobolev orthogonal polynomials

Abstract

We study properties of the monic polynomials $\{Q_n\}_{n\in\bbfN}$ orthogonal with respect to the Sobolev inner product $$(p,q)_S= \int infty_0 (p,p')\pmatrix 1 & \mu\\ \mu &\lambda\endpmatrix \pmatrix q\\ q'\endpmatrix x alpha e -x} dx,$$ where $\lambda- \mu > 0$ and $\alpha> -1$. This inner product can be expressed as $$(p,q)_S= \int infty_0 p(x) q(x)((\mu+ 1) x- \alpha\mu) x \alpha- 1} e -x} dx+ \lambda\int infty_0 p'q' x alpha e -x} dx,$$ when $\alpha> 0$. In this way, the measure which appears in the first integral is not positive on $[0,\infty)$ for $\mu\in \bbfR\setminus[- 1,0]$. The aim of this paper is the study of analytic properties of the polynomials $Q_n$. First, we give an explicit representation for $Q_n$ using an algebraic relation between Sobolev and Laguerre polynomials together with a recursive relation for $\widetilde k_n= (Q_n,Q_n)_S$. Then we consider analytic aspects. We first establish the strong asymptotics of $Q_n$ on $\bbfC\setminus[0,\infty)$ when $\mu\in \bbfR$ and we also obtain an asymptotic expression on the oscillatory region, that is, on $(0,\infty)$. Then we study the Plancherel-Rotach asymptotics for the Sobolev polynomials $Q_n(nx)$ on $\bbfC\setminus[0, 4]$ when $\mu\in (- 1,0]$. As a consequence of these results we obtain the accumulation sets of zeros and of the scaled zeros of $Q_n$. We also give a Mehler-Heine type formula for the Sobolev polynomials which is valid on compact subsets of $\bbfC$ when $\mu\in (-1,0]$, and hence in this situation we obtain a more precise result about the asymptotic behaviour of the small zeros of $Q_n$. This result is illustrated with three numerical examples.