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

In this paper the discrete Sobolev inner product 〈p, q〉 = ∫ p(x)q(x)dμ+ r ∑ i=0 Mi p (c)q(c) is considered, whereμ 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. © 2015 Elsevier Inc. All rights reserved.