On some analytic properties of slice poly-regular Hermite polynomials

TL;DR

This paper explores the properties of quaternionic Hermite polynomials, establishing their integral representations, orthogonality, generating functions, and identities, thus extending classical Hermite polynomial theory into the quaternionic setting.

Contribution

It introduces a quaternionic analogue of complex Hermite polynomials and analyzes their analytic properties, including orthogonality and generating functions, which is a novel extension of classical polynomial theory.

Findings

They form an orthogonal basis in quaternionic Hilbert spaces.

Integral and operational formulas for the polynomials are derived.

Various generating functions and identities, including quadratic recurrence formulas, are established.

Abstract

We consider a quaternionic analogue of the univariate complex Hermite polynomials and study some of their analytic properties in some detail. We obtain their integral representation as well as the operational formulas of exponential and Burchnall types they obey. We show that they form an orthogonal basis of both the slice and the full Hilbert spaces on the quaternions with respect to the Gaussian measure. We also provide different types of generating functions. Remarkably identities, including quadratic recurrence formulas of Nielsen type are derived.