Strong and Plancherel–Rotach Asymptotics of Non-diagonal Laguerre–Sobolev Orthogonal Polynomials Original Research Article

We study properties of the monic polynomials {Qn}n∈N orthogonal with respect to the Sobolev inner product(p, q)S=∫∞0 (p, p′) 1μ μλqq′ xαe−x dx,where λ−μ2>0 and α>−1. This inner product can be expressed as(p, q)S=∫∞0 p(x) q(x)((μ+1) x−αμ) xα−1e−x dx+λ ∫∞0 p′q′xαe−x dx,when α>0. In this way, the measure which appears in the first integral is not positive on [0, ∞) for μ∈R\[−1, 0]. The aim of this paper is the study of analytic properties of the polynomials Qn. First we give an explicit representation for Qn using an algebraic relation between Sobolev and Laguerre polynomials together with a recursive relation for kn=(Qn, Qn)S. Then we consider analytic aspects. We first establish the strong asymptotics of Qn on C\[0, ∞) when μ∈R and we also obtain an asymptotic expression on the oscillatory region, that is, on (0, ∞). Then we study the Plancherel–Rotach asymptotics for the Sobolev polynomials Qn(nx) on C\[0, 4] when μ∈(−1, 0]. As a consequence of these results we obtain the accumulation sets of zeros and of the scaled zeros of Qn. We also give a Mehler–Heine type formula for the Sobolev polynomials which is valid on compact subsets of C when μ∈(−1, 0], and hence in this situation we obtain a more precise result about the asymptotic behaviour of the small zeros of Qn. This result is illustrated with three numerical examples.