Connection formulas for general discrete Sobolev polynomials. Mehler-Heine asymptotics

TL;DR

This paper derives connection formulas for discrete Sobolev orthogonal polynomials and establishes Mehler-Heine asymptotics at hard edges of the support, including symmetric measures, with various examples.

Contribution

It introduces new connection formulas for Sobolev orthogonal polynomials and extends Mehler-Heine asymptotics to cases involving hard edges of the measure support.

Findings

Connection formulas for Sobolev orthogonal polynomials are established.

Mehler-Heine asymptotics are derived at hard edges of the support.

Symmetric measure cases are specifically analyzed.

Abstract

In this paper the discrete Sobolev inner product $$< p,q > =\int p(x) q(x) \,d\mu + \sum_{i=0}^r M_i \, p^{(i)}(c) \, q^{(i)}(c)$$ is considered, where $\mu$ is a finite positive Borel measure supported on an infinite subset of the real line, $c\in\mathbb{R}$ and $\, M_i \ge 0, \, i = 0, 1, ..., r.$ Connection formulas for the orthonormal polynomials associated with $< ., . >$ are obtained. As a consequence, for a wide class of measures $\mu$, we give the Mehler-Heine asymptotics in the case of the point $c$ is a hard edge of the support of $\mu$. In particular, the case of a symmetric measure $\mu$ is analyzed. Finally, some examples are presented.