_{1}

^{*}

In this comment we will demonstrate that one of the main formulas given in Ref. [ 1 ] is incorrect.

It is well known that for a family of orthogonal polynomials the so-called “generating functions” corresponding to this class of functions are a useful tool for their study, see [2,3]. Usually, a generating function is a function of two variables, analytic in some set, so that

For example, we have the following generating function of Hermite polynomials, because we can write:

Note that it is important to specify the subset where the function is well defined and analytic. For example, for Legendre polynomials we have

where it is important to specify the domain of the variables, because, in other case, for example with the choise, formula (1) is meaningless.

The extension to the matrix framework for the classical case of Gegenbauer [

• For a matrix such that, , i.e, A is say positive stable matrix, the Hermite matrix polynomials sequence is defined by the generating function [

•

• For a matrix such that for every integer, and is a complex number with, the Laguerre matrix polynomials sequence is defined by the generating function [

Recently, in Ref. [

where is a positive stable matrix, i.e., satisfies for all eigenvalue, and m is a positive integer. This Formula (7) turns out to be the key for the development of the properties mentioned in the paper [

where is the exponential matrix. Of course, has sense only for. Thus, Expression (7) is meaningless if the term is zero. Then, we only need to consider, for example, , and and with this choice we have. Thus, (7) is meaningless.

Therefore, I ask the authors of Ref. [