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

We study properties of the monic polynomials {Q(n)}(n is an element ofN) orthogonal with respect to the Sobolev inner product [GRAPHICS] where lambda-mu (2)>0 and alpha> - 1. This inner product can be expressed as (p, q) s = integral (infinity)(o) p(x)q(x)(mu +1) x - alpha mu) x(alpha -1)e(-x) dx+lambda integral (infinity)(o) 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, infinity) for mu is an element of R\ [- 1, 0]. The aim of this pal,er 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 (k) over tilde (n) = (Q(n), Q(n))(S). Then we consider analytic aspects. We first establish the strong asymptotics of Q(n) on C\[0, infinity) when mu is an element ofR and we also obtain an asymptotic expression on the oscillatory region. that is. on (0, infinity). Then we study the Plancherel-Rotach asymptotics for the Sobolev polynomials Q(n)(nx) on C\[0, 4] when mu is an element of(-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 C when mu is an element of(-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. (C) 2001 Academic Press.