Connection formulas for general discrete Sobolev polynomials: Mehler–Heine asymptotics

摘要

In this paper the discrete Sobolev inner product 〈p,q〉=∫p(x)q(x)dμ+∑i=0rMip(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∈R and  Mi ≥ 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.