The combinatorics of associated Hermite polynomials

TL;DR

This paper introduces a combinatorial framework for associated Hermite polynomials, establishing their orthogonality and providing multiple combinatorial interpretations and identities related to their moments.

Contribution

It develops a novel combinatorial model for associated Hermite polynomials, proving orthogonality and deriving various identities and interpretations.

Findings

Orthogonality proven via sign-reversing involution

Multiple combinatorial interpretations of moments provided

Derived identities and generating functions for the polynomials

Abstract

We develop a combinatorial model of the associated Hermite polynomials and their moments, and prove their orthogonality with a sign-reversing involution. We find combinatorial interpretations of the moments as complete matchings, connected complete matchings, oscillating tableaux, and rooted maps and show weight-preserving bijections between these objects. Several identities, linearization formulas, the moment generating function, and a second combinatorial model are also derived.