Asymptotic properties of generalized Laguerre orthogonal polynomials

In the present paper we deal with the polynomials L-n((a,M,N)) (x) orthogonal with respect to the Sobolev inner product (p,q) = 1/Gamma(alpha+1) integral(0)(infinity)p(x)q(x)x(alpha)e(-x)dx + Mp(0)q(0) + Np'(0)q'(0), N, M greater than or equal to (0), alpha > -1, firstly introduced by Koekoek and Meijer in 1993 and extensively studied in the last years. We present some new asymptotic properties of these polynomials and also a limit relation between the zeros of these polynomials and the zeros of Bessel function J(alpha)(x). The results are illustrated with numerical examples. Also, some general asymptotic formulas for generalizations of these polynomials are conjectured.