Asymptotics for a generalization of Hermite polynomials

2009/09/03 by M. Alfaro, Manuel Alfaro, Juan J. Moreno–Balcázar +9
Mathematics · Physics and Astronomy · #33C45 #42C05 #Advanced Mathematical Theories and Applications #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Mathematics and Applications #math.CA #msc:33C45 #msc:42C05

paper · pdf · doi:10.48550/arxiv.0909.0617

18 pages

arxiv created 2009/09/03 · openalex publication_date 2009/09/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler--Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Hermite polynomials towards the origin.

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