PACIFIC JOURNAL OF MATHEMATICS Vol. 257, No. 1, 2012 ON ORTHOGONAL POLYNOMIALS WITH RESPECT TO CERTAIN DISCRETE SOBOLEV INNER PRODUCT FRANCISCO MARCELLÁN,RAMADAN ZEJNULLAHU, BUJAR FEJZULLAHU AND EDMUNDO HUERTAS In this paper we deal with sequences of polynomials orthogonal with respect to the discrete Sobolev inner product 〈 f, g〉 S =  ∞ 0 ω( x ) f ( x ) g( x ) dx + Mf (ξ) g(ξ) + Nf ′ (ξ) g ′ (ξ ), where ω is a weight function, ξ ≤ 0, and M, N ≥ 0. The location of the zeros of discrete Sobolev orthogonal polynomials is given in terms of the ze- ros of standard polynomials orthogonal with respect to the weight function ω. In particular, for ω( x ) = x α e −x we obtain the asymptotics for discrete Laguerre–Sobolev orthogonal polynomials. 1. Introduction Polynomials orthogonal with respect to an inner product (1) 〈 f , g〉=  E ω(x ) f (x )g(x ) dx + Mf (ξ)g(ξ) + Nf ′ (ξ)g ′ (ξ ), where ξ is a real number and d µ is a positive Borel measure supported on an infinite subset E of the real line have been considered by several authors (see, for instance, [Alfaro et al. 1992; López et al. 1995; Marcellán and Ronveaux 1990; Marcellán and Van Assche 1993] and the references therein). They are known in the literature as Sobolev-type or discrete Sobolev orthogonal polynomials. Special attention has been paid to their algebraic and analytic properties of these polynomials, in particular, the distribution of their zeros taking into account the location of the point ξ with respect to the set E . When E is the interval [0, +∞) and ξ = 0, Meijer [1993a] analyzed some analytic properties of the zeros of the so called discrete Sobolev orthogonal poly- nomials (1). Some results of [Meijer 1993a] are direct generalizations of the re- sults of [Koekoek and Meijer 1993], where the weight function is the Laguerre MSC2010: primary 33C47; secondary 42C05. Keywords: orthogonal polynomials, discrete Sobolev polynomials, Laguerre polynomials, asymptotics. 167