Old and New Results About Relativistic Hermite Polynomials

TL;DR

This paper reviews and extends the theory of relativistic Hermite polynomials by providing new proofs and results, utilizing moments, subordination, and Nagel's identity, connecting them to Gegenbauer polynomials.

Contribution

It offers new proofs and novel results about relativistic Hermite polynomials, enhancing understanding of their properties and relations to classical polynomials.

Findings

New proofs of known properties of RHP

New results about RHP properties

Connections between RHP and Gegenbauer polynomials

Abstract

The relativistic Hermite polynomials (RHP) were introduced in 1991 by Aldaya et al. in a generalization of the theory of the quantum harmonic oscillator to the relativistic context. These polynomials were later related to the more classical Gegenbauer (or ultraspherical) polynomials in a study by Nagel. Thus some of their properties can be deduced from the properties of the well-known Gegenbauer polynomials, as underlined by M. Ismail. In this report we give new proofs of already known results but also new results about these polynomials. We use essentially three basic tools: the representation of polynomials as moments, the subordination tool and Nagel's identity.