A new approach to the asymptotics of Sobolev type orthogonal polynomials

Abstract

This paper deals with Mehler-Heine type asymptotic formulas for the so-called discrete Sobolev orthogonal polynomials whose continuous part is given by Laguerre and generalized Hermite measures. We use a new approach which allows to solve the problem when the discrete part contains an arbitrary (finite) number of mass points.

On Sobolev orthogonal polynomials

“…uniformly on compact subsets of the exterior of the real positive semiaxis. When c = 0 and A is a non-singular diagonal matrix of size m + 1, the following asymptotic properties of the Sobolev orthogonal polynomials with respect to the inner product (9.1) were obtained in [7], It should be pointed out that the above Mehler-Heine formula cannot be directly deduced from the connection formula (7.6). As an interesting consequence of the Mehler-Heine formula and the Hurwitz Theorem, the local behavior of zeros of these Sobolev orthogonal polynomials can be deduced.…”

Zeros of varying Laguerre–Krall orthogonal polynomials

“…The Mehler-Heine type asymptotic formula for these polynomials was obtained in [3] together with other asymptotic results which have been extended very recently to Sobolev type orthogonal polynomials [2]. Denoting L…”

Connection formulas for general discrete Sobolev polynomials: Mehler–Heine asymptotics

“…The result is a simple consequence of Theorem 1 and the symmetrization process described above. Indeed, from (27) and since p n (x) are symmetric polynomials, we get u n (0)u [2] n−1 (0) . .…”

“…However, the situation is quite different in the case of measures with unbounded support. So, in [2], it was shown that this connection formula is not the adequate to study neither relative asymptotics nor Mehler-Heine formula when µ is the Laguerre weight.…”