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.