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) or thogonal with respect to the Sobolev inner product [GRAPHICS] where lambda-mu (2)>0 and alpha> - 1. This inner product can be expressed a s (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 r elation between Sobolev and Laguerre polynomials together with a recursive relation For (k) over tilde (n) = (Q(n), Q(n))(S). Then we consider analyti c aspects. We first establish the strong asymptotics of Q(n) on C\[0, infin ity) 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 Pla ncherel-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 obta in 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 o f the small zeros of Q(n). This result is illustrated with three numerical examples. (C) 2001 Academic Press.