A new type of Hermite matrix polynomial series

A new type of Hermite matrix polynomial series (E. DEFEZ, M. TUNG), Quaestiones Mathematicae, 41(2), 2018, 205–212.

Abstract: Conventional Hermite polynomials emerge in a great diversity of applications
in mathematical physics, engineering, and related fields. However, in physical
systems with higher degrees of freedom it will be of practical interest to extend
the scalar Hermite functions to their matrix analogue. This work introduces various
new generating functions for Hermite matrix polynomials and examines existence and
convergence of their associated series expansion by using Mehlerís formula for the
general matrix case. Moreover, we derive interesting new relations for even- and
odd-power summation in the generating-function expansion containing Hermite matrix
polynomials. Some new results for the scalar case are also presented.