Question about integral of product of four Hermite polynomials integrated with squared weight

TL;DR

This paper investigates the integral of four Hermite polynomials with squared weight, revealing recursive properties and posing open questions about explicit formulas, with connections to Hermitian matrices.

Contribution

It uncovers recursive structures in integrals of four Hermite polynomials and raises questions about explicit formulas and matrix interpretations.

Findings

Integral generates symmetric polynomials with recursive properties

Open question on explicit formula for the integral

Connection to Hermitian matrix interpretation

Abstract

We found that the integral of four Hermite polynomials integrated with squared weight over the real line generates symmetric polynomials with a beautiful recursive property. We pose a question whether that integral admit an explicit formula or not. The question has a certain interpretation in terms of Hermitian matrices.