Δ-Sobolev orthogonal polynomials of Meixner type:: asymptotics and limit relation

Let {Q(n) (x)}(n) be the sequence of monic polynomials orthogonal with respect to the Sobolev-type inner product < p(x), r(x)>(S) = < u(0), p(x)r(x)> + lambda < u(1), (Delta p)(x)(Delta r)(x)>, where, lambda >= 0, (Delta f)(x) = f (x + 1) - f (x) denotes the forward difference operator and (u(0), u(1)) is a Delta-coherent pair of positive-definite linear functionals being u(1) the Meixner linear functional. In this paper, relative asymptotics for the {Q(n)(x)}(n) sequence with respect to Meixner polynomials on compact subsets of C\(0, +infinity) is obtained. This relative asymptotics is also given for the scaled polynomials. In both cases, we deduce the same asymptotics as we have for the self-A-coherent pair, that is, when u(0) = u(1) is the Meixner linear functional. Furthermore, we establish a limit relation between these orthogonal polynomials and the Laguerre-Sobolev orthogonal polynomials which is analogous to the one existing between Meixner and Laguerre polynomials in the Askey scheme. (c) 2004 Elsevier B.V. All rights reserved.