Old and New Results About Relativistic Hermite Polynomials

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.