A Comment on “On Humbert Matrix Polynomials of Two Variables”

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

• Humbert Matrix Polynomials

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

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):

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].

References

1. G. S. Khammash and A. Shehata, “On Humbert Matrix Polynomials of Two Variables,” Advances in Pure Mathematics, Vol. 2, No. 6, 2012, pp. 423-427.

2. T. S. Chihara, “An Introduction to Orthogonal Polynomials,” Gordon and Breach, New York, 1978.

3. N. N. Lebedev, “Special Functions and Their Applications,” 2nd Edition, Dover Pub-lications, INC., New York, 1972.

4. L. Jódar, R. Company and E. E. Ponsoda, “Orthogonal Matrix Polynomials and Systems of Second Order Differential Equations,” Differential Equations and Dynamical Systems, Vol. 3, No. 3, 1995, pp. 269-288.

5. L. Jódar, R. Company and E. Navarro, “Laguerre Matrix Polynomials and Systems of Second Order Differential Equations,” Applied Numeric Mathematics, Vol. 15, No. 1, 1994, pp. 53-64. doi:10.1016/0168-9274(94)00012-3

6. L. Jódar and R. Company, “Hermite Matrix Polynomials and Second Order Matrix Differential Equations,” Journal of Approximation Theory and its Applications, Vol. 12, No. 2, 1996, pp. 20-30.

7. E. Defez, L. Jódar and A. G. Law, “Jacobi Matrix Differential Equation, Polynomial Solutions, and Their Properties,” Computers and Mathematics with Applications, Vol. 48, No. 5-6, 2004, pp. 789-803.

8. E. Defez and L. Jódar, “Chebyshev Matrix Polynomials and Second Order Matrix Differential Equations,” Utilitas Mathematica, Vol. 61, 2002, pp. 107-123.

9. E. Defez and L. Jódar, “Some Applications of the Hermite Matrix Polynomials Series Expansions,” Journal of Computational and Applied Mathematics, Vol. 99, No. 1-2, 1998, pp. 105-117. doi:10.1016/S0377-0427(98)00149-6

10. E. Defez, M. M. Tung and J. Sastre, “Improvement on the Bound of Hermite Matrix Polynomials,” Linear Algebra and Its Applications, Vol. 434, No. 8, 2011, pp. 1910-1919. doi:10.1016/j.laa.2010.12.015

11. J. Sastre, J. J. Ibánez, E. Defez and P. Ruiz, “Efficient Orthogonal Matrix Polynomial Based Method for Computing Matrix Exponential,” Applied Mathematics and Computation, Vol. 217, No. 14, 2011, pp. 6451-6463. doi:10.1016/j.amc.2011.01.004

12. J. Sastre, J. J. Ibánez, E. Defez and P. Ruiz, “Computing Matrix Functions Solving Coupled Differential Models,” Mathematical and Computer Modelling, Vol. 50, No. 5-6, 2009, pp. 831-839. doi:10.1016/j.mcm.2009.05.012

13. A. Durán and F. A. Grünbaum, “A Survey on Orthogonal Matrix Polynomials Satisfying Second Order Differential Equations,” Journal of Computational and Applied Mathematics, Vol. 178, No. 1-2, 2005, pp. 169-190. doi:10.1016/j.cam.2004.05.023