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Connection formulas for general discrete Sobolev polynomials: Mehler-Heine asymptotics. (English) Zbl 1410.42026
Summary: In this paper the discrete Sobolev inner product $\langle p, q \rangle = \int p(x) q(x) d \mu + \sum_{i = 0}^r M_i p^{(i)}(c) q^{(i)}(c)$ is considered, where $$\mu$$ is a finite positive Borel measure supported on an infinite subset of the real line, $$c \in \mathbb{R}$$ and $$M_{i} 0$$, $$i = 0, 1,\dots, r$$. Connection formulas for the orthonormal polynomials associated with $$\langle \dot , dot\rangle$$ 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.

##### MSC:
 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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