Zero sets of bivariate Hermite polynomials

Author(s)

Area, Ivan ^[1] ; Dimitrov, Dimitar K. ^[2] ; Godoy, Eduardo ^[3]

Affiliation

^[1] Univ Vigo, Dept Matemat Aplicada 2, EE Telecomunicac, Vigo 36310 - Spain

^[2] Univ Estadual Paulista, IBILCE, Dept Matemat Aplicada, BR-15054000 Sao Jose Do Rio Preto, SP - Brazil

^[3] Univ Vigo, Dept Matemat Aplicada 2, EE Ind, Vigo 36310 - Spain

• Document type: Journal article
• Source: Journal of Mathematical Analysis and Applications; v. 421, n. 1, p. 830-841, JAN 1 2015.
• Web of Science Citations: 2

Abstract

We establish various properties for the zero sets of three families of bivariate Hermite polynomials. Special emphasis is given to those bivariate orthogonal polynomials introduced by Hermite by means of a Rodrigues type formula related to a general positive definite quadratic form. For this family we prove that the zero set of the polynomial of total degree n + m consists of exactly n + m disjoint branches and possesses n + m asymptotes. A natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property. The results show that the properties of the zero sets, considered as affine algebraic curves in R-2, are completely different for the three families analyzed. (c) 2014 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 09/13832-9 - Orthogonal polynomials, special functions and applications
• Grantee: Dimitar Kolev Dimitrov
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 13/23606-1 - Methods for approximate calculus of sums and series
• Grantee: Dimitar Kolev Dimitrov
• Support Opportunities: Research Grants - Visiting Researcher Grant - International