1. Introduction and Motivation
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
(1)
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 [4], Laguerre [5], Hermite [6], Jacobi [7] and Chebyshev [8] polynomials has been made in recent years, and properties and applications of different classes for these matrix polynomials are given in several papers, see [9-13] for example. The importance of the generating function for orthogonal matrix polynomials is similar to the scalar case, taking into account the possible additional spectral restrictions (for a matrix 
we will denote by 
the spectrum set
). For example:
• For a matrix 
such that
, 
, i.e, A is say positive stable matrix, the Hermite matrix polynomials sequence 
is defined by the generating function [6]:
• 
• 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 [5]:
2. The Detected Error
Recently, in Ref. [1], the Humbert matrix polynomials of two variables are defined using the generating matrix function given in Formula (7):
(7)
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 [1]. However, we will see that Formula (7) is incorrect. For this, first we have to observe that for a matrix A, we define
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. [1] to clarify the domain of choice for the variables t, s in Formula (7) in order to guarantee the validity of the remaining formulas which are derived from (7) and are used in the remainder of [1].