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

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.

26 pages

References in corpus (2)

• Mehler-Heine asymptotics for multiple orthogonal polynomials
• Discrete Laguerre-Sobolev expansions: A Cohen type inequality