A new type of Hermite matrix polynomial series

[EN] 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 Mehler¿s 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.The authors thank the Spanish Ministerio de Economia y Competitividad and the European Regional Development Fund (ERDF) for financial support under grant TIN2014-59294-P.Defez Candel, E.; Tung, MM. (2018). A new type of Hermite matrix polynomial series. Quaestiones Mathematicae. 41(2):205-212. https://doi.org/10.2989/16073606.2017.1376231S20521241

On Hermite-Hermite matrix polynomials

summary:In this paper the definition of Hermite-Hermite matrix polynomials is introduced starting from the Hermite matrix polynomials. An explicit representation, a matrix recurrence relation for the Hermite-Hermite matrix polynomials are given and differential equations satisfied by them is presented. A new expansion of the matrix exponential for a wide class of matrices in terms of Hermite-Hermite matrix polynomials is proposed