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

Abstract

a b s t r a c tIn this paper the discrete Sobolev inner productis considered, where μ is a finite positive Borel measure supported on an infinite subset of the real line, c ∈ R and M i ࣙ 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.

Differential operator for discrete Gegenbauer–Sobolev orthogonal polynomials: Eigenvalues and asymptotics

“…In the context of Sobolev orthogonality there is a wide literature, we can cite the surveys [14] and [16], and the references therein. Even more recently and conceptually closer to the inner product (1) we can point out [12,13,18] among others.…”

Asymptotic behavior of varying discrete Jacobi–Sobolev orthogonal polynomials

“…1]. The idea is that the coefficients b i (n) in (12) can be obtained as a solution of a homogeneous linear system of j + 1 equations and j + 2 unknowns. In our concrete case, we can compute explicitly the entries of the corresponding coefficient matrix.…”

“…The following result gives us this expansion. In a more general framework it has been established in [12,Th. 1].…”

“…Acknowledgements: We thank the two anonymous referees for their useful suggestions to improve the paper. In special, one of the referees provided us with some references such as [12] which have been relevant for our research.…”

“…We remark that the techniques used in [6] are not useful in this case, and now we need to use more powerful techniques based on those considered in [11]. More recently, in [12] the same authors have even improved these techniques in such a way that they have obtained relevant results for the orthogonal polynomials with respect to a non-varying discrete Sobolev inner product being µ 0 a general measure.…”