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 the slice Hilbert space L-2(L-I; e(-vertical bar q vertical bar 2) d lambda(I)) of all quaternionic-valued functions defined the whole quaternions space and subject to norm boundedness with respect to the Gaussian measure on a given slice as well as of the full left quaternionic Hilbert space L-2(H; e(-vertical bar q vertical bar 2) d lambda) of square integrable functions on quaternions with respect to the Gaussian measure on the whole H equivalent to R-4. We also provide different types of generating functions. Remarkable identities, including quadratic recurrence formulas of Nielsen type, are also derived.