The Expansion of Wronskian Hermite Polynomials in the Hermite Basis

TL;DR

This paper provides explicit formulas for expressing Wronskian Hermite polynomials in the Hermite basis, derives bounds on their roots, and explores implications for irreducibility and generalizations to broader polynomial classes.

Contribution

It introduces explicit coefficient formulas, root bounds, and extends results to a wider class of polynomials, advancing understanding of Wronskian Hermite polynomials.

Findings

Explicit formula for coefficients in Hermite basis

Upper bounds for roots of Wronskian Hermite polynomials

Generalization to larger classes of polynomials

Abstract

We express Wronskian Hermite polynomials in the Hermite basis and obtain an explicit formula for the coefficients. From this we deduce an upper bound for the modulus of the roots in the case of partitions of length 2. We also derive a general upper bound for the modulus of the real and purely imaginary roots. These bounds are very useful in the study of irreducibility of Wronskian Hermite polynomials. Additionally, we generalize some of our results to a larger class of polynomials.