Asymptotic behavior of varying discrete Jacobi–Sobolev orthogonal polynomials

Abstract

In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler-Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev orthogonal polynomials and the Jacobi ones. Moreover, this asymptotic behavior provides an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other well-known special functions. We generalize some results appeared in the literature very recently.

Eigenvalue Problem for Discrete Jacobi–Sobolev Orthogonal Polynomials

“…(13a) This case has been proved in ( [33], Lemma 1). (13b) We use the same technique as in ( [33], Lemma 1), getting…”

“…For our purposes, we need to know some asymptotic behaviors of these kernel polynomials evaluated at the points 1 and −1. Next, we introduce a result which generalizes some particular cases obtained in ( [33], Lemma 1) or ( [35], Pag. 147).…”

“…Finally, it is enough to apply these results together with (29) and (30) in (33) to deduce the result.…”

Differential operator for discrete Gegenbauer–Sobolev orthogonal polynomials: Eigenvalues and asymptotics

“…In the context of Sobolev orthogonality there is a wide literature, we can cite the surveys [14] and [16], and the references therein. Even more recently and conceptually closer to the inner product (1) we can point out [12,13,18] among others.…”

“…As we have commented in the previous proposition, this result was also established in a similar context in [18, Th. 1] (see also [12,Th. 2]).…”