Zeros of exceptional Hermite polynomials

• Published in: Journal of approximation theory Vol. 200; pp. 28 - 39
• Main Authors: Kuijlaars, A.B.J., Milson, R.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.12.2015
• Subjects: Exceptional orthogonal polynomials; Hermite polynomials; Zero distribution; Zero distribution; Hermite polynomials; Exceptional orthogonal polynomials
• ISSN: 0021-9045; 1096-0430
• DOI: 10.1016/j.jat.2015.07.002

We study the zeros of exceptional Hermite polynomials associated with an even partition λ. We prove several conjectures regarding the asymptotic behaviour of both the regular (real) and the exceptional (complex) zeros. The real zeros are distributed as the zeros of usual Hermite polynomials and, after contracting by a factor 2n, we prove that they follow the semi-circle law. The non-real zeros tend to the zeros of the generalized Hermite polynomial Hλ, provided that these zeros are simple. It was conjectured by Veselov that the zeros of generalized Hermite polynomials are always simple, except possibly for the zero at the origin, but this conjecture remains open.