an:06890741
Zbl 1410.42026
Pe??a, A.; Rezola, M. L.
Connection formulas for general discrete Sobolev polynomials: Mehler-Heine asymptotics
EN
Appl. Math. Comput. 261, 216-230 (2015).
00404123
2015
j
42C05 33C45
discrete Sobolev polynomials; connection formulas; Mehler-Heine type formulas; Bessel functions; asymptotic zero distribution
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.