On some analytic properties of slice poly-regular Hermite polynomials

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.