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

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DOI: 10.4236/apm.2013.35066 3,735 Downloads 5,627 Views Citations

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Vicente Soler Basauri

Affiliation(s)

Departamento de Matemática Aplicada, Universitat Politècnica de València, Valencia, Spain.

ABSTRACT

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

KEYWORDS

Humbert Matrix Polynomials

Cite this paper

V. Basauri, "A Comment on “On Humbert Matrix Polynomials of Two Variables”," Advances in Pure Mathematics, Vol. 3 No. 5, 2013, pp. 470-471. doi: 10.4236/apm.2013.35066.

Conflicts of Interest

The authors declare no conflicts of interest.

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

[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