Alternate CM-pair-based proof of orthogonality for Exceptional Hermite Polynomials

Develop an elementary proof of the orthogonality of the family of λ-Exceptional Hermite Polynomials {H_n^{(λ)}(x,y) : n ∈ I^{(λ)}} under the Hermite-like inner product defined by integration against the weight exp(x^2/(4y)) divided by τ^{(λ)}(x,y)^2 (for the appropriate choices of λ and y), using only the Calogero–Moser pair representation (X^λ, Z^λ) and the expansion formula expressing H_n^{(λ)} as linear combinations of classical Hermite polynomials (as given by Theorem 4), together with elementary linear algebra and basic properties of integrals, avoiding advanced analytic techniques.

Background

The paper establishes a precise correspondence between Exceptional Hermite Polynomials (XHPs) and Calogero–Moser (CM) pairs, including explicit CM-pair formulas for the generating function and an expansion of XHPs in terms of classical Hermite polynomials. Orthogonality of XHPs under a Hermite-like inner product is known from prior work, typically proved using analytic methods.

The authors aimed to leverage their CM-pair framework—particularly the explicit expansion (Theorem 4)—to give a more elementary proof of orthogonality relying on linear algebra and integrals. They explicitly state they were unable to produce such a proof, leaving the existence of an elementary CM-based orthogonality proof as an open task.