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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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