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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.

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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