Zeros of exceptional Hermite polynomials

TL;DR

This paper investigates the asymptotic distribution of zeros of exceptional Hermite polynomials, showing real zeros follow the semi-circle law and non-real zeros tend to zeros of generalized Hermite polynomials, confirming several conjectures.

Contribution

It proves conjectures about the asymptotic behavior of zeros of exceptional Hermite polynomials, including their distribution and relation to generalized Hermite polynomial zeros.

Findings

Real zeros follow the semi-circle law after scaling.

Non-real zeros tend to zeros of generalized Hermite polynomials.

Several conjectures about zeros' behavior are confirmed.

Abstract

We study the zeros of exceptional Hermite polynomials associated with an even partition $\lambda$. We prove several conjectures regarding the asymptotic behavior 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 $\sqrt{2n}$, we prove that they follow the semi-circle law. The non-real zeros tend to the zeros of the generalized Hermite polynomial $H_{\lambda}$, 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.