Durfee rectangles and pseudo-Wronskian equivalences for Hermite polynomials

TL;DR

This paper explores the equivalence of various Darboux transformations on the harmonic oscillator, deriving Hermite polynomial identities, and applying these to optimize representations of exceptional Hermite polynomials and Painlevé IV solutions.

Contribution

It introduces a new framework linking Darboux transformations, Hermite polynomial identities, and combinatorial structures, optimizing the transformation process and representations of special functions.

Findings

Derived new Hermite polynomial determinant identities.

Characterized the equivalence classes of Darboux transformations.

Provided minimal order determinants for efficient transformations.

Abstract

We study an equivalence class of iterated rational Darboux transformations applied on the harmonic oscillator, showing that many choices of state adding and state deleting transformations lead to the same transformed potential. As a by-product, we derive new identities between determinants whose entries are Hermite polynomials. These identities have a combinatorial interpretation in terms of Maya diagrams, partitions and Durfee rectangles, and serve to characterize the equivalence class of rational Darboux transformations. Since the determinants have different orders, we analyze the problem of finding the minimal order determinant in each equivalence class, or equivalently, the minimum number of Darboux transformations. The solution to this problem has an elegan graphical interpretation. The results are applied to provide alternative and more efficient representations for exceptional Hermite polynomials and rational solutions of the Painlev\'e IV equation.