Coefficients of Wronskian Hermite polynomials

TL;DR

This paper derives explicit formulas for coefficients of Wronskian Hermite polynomials using combinatorial methods, explores their asymptotic behavior, and generalizes classical polynomial correspondences and zeros distribution.

Contribution

It introduces combinatorial expressions for coefficients, analyzes asymptotics, and extends polynomial correspondences to Wronskian Hermite and Laguerre polynomials.

Findings

Coefficients expressed via symmetric group characters and hook lengths.

Asymptotic behavior when core length tends to infinity.

Generalization of Hermite-Laguerre polynomial correspondence.

Abstract

We study Wronskians of Hermite polynomials labelled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the characters of irreducible representations of the symmetric group, and also in terms of hook lengths. Further, we derive the asymptotic behaviour of the Wronskian Hermite polynomials when the length of the core tends to infinity, while fixing the quotient. Via this combinatorial setting, we obtain in a natural way the generalization of the correspondence between Hermite and Laguerre polynomials to Wronskian Hermite polynomials and Wronskians involving Laguerre polynomials. Lastly, we generalize most of our results to polynomials that have zeros on the $p$-star.