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

In this paper the discrete Sobolev inner product $< p,q > =\int p(x) q(x) \,dμ+ \sum_{i=0}^r M_i \, p^{(i)}(c) \, q^{(i)}(c)$ is considered, where $μ$ 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 $μ$, we give the Mehler-Heine asymptotics in the case of the point $c$ is a hard edge of the support of $μ$. In particular, the case of a symmetric measure $μ$ is analyzed. Finally, some examples are presented.

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