On Humbert Matrix Polynomials of Two Variables
Affiliation(s)
Department of Mathematics, Al-Aqsa University, Gaza Strip, Palestine. Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt.
Department of Mathematics, Al-Aqsa University, Gaza Strip, Palestine. Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt.
ABSTRACT
In this paper we introduce Humbert matrix polynomials of two variables. Some hypergeometric matrix representations of the Humbert matrix polynomials of two variables, the double generating matrix functions and expansions of the Humbert matrix polynomials of two variables in series of Hermite polynomials are given. Results of Gegenbauer ma-trix polynomials of two variables follow as particular cases of Humbert matrix polynomials of two variables.
In this paper we introduce Humbert matrix polynomials of two variables. Some hypergeometric matrix representations of the Humbert matrix polynomials of two variables, the double generating matrix functions and expansions of the Humbert matrix polynomials of two variables in series of Hermite polynomials are given. Results of Gegenbauer ma-trix polynomials of two variables follow as particular cases of Humbert matrix polynomials of two variables.
Cite this paper
G. Khammash and A. Shehata, "On Humbert Matrix Polynomials of Two Variables," Advances in Pure Mathematics, Vol. 2 No. 6, 2012, pp. 423-427. doi: 10.4236/apm.2012.26064.
G. Khammash and A. Shehata, "On Humbert Matrix Polynomials of Two Variables," Advances in Pure Mathematics, Vol. 2 No. 6, 2012, pp. 423-427. doi: 10.4236/apm.2012.26064.
References
[1] A. G. Constantine and R. J. Mairhead, “Partial Differential Equations for Hypergeometric Functions of Two Argument Matrices,” Journal of Multivariate Analysis, Vol. 2, No. 3, 1972, pp. 332-338. doi:10.1016/0047-259X(72)90020-6. [2] R. J. Muirhead, “Systems of Partial Differential Equations for Hypergeometric Functions of Matrix Argument,” Annals of Mathematical Statistics, Vol. 40, No. 3, 1970, pp. 991-1001. doi:10.1214/aoms/1177696975 [3] A. Terras, “Special Functions the Symmetric Space of Positive Matrices,” SIAM Journal on Mathematical Analysis, Vol. 16, No. 3, 1985, pp. 620-640. doi:10.1137/0516046 [4] A. T. James, “Special Functions of Matrix and Single Argument in Statistics in Theory and Application of Special Functions,” Academic Press, New York, 1975. [5] A. J. Duran, “Markov’s Theorem for Orthogonal Matrix Polynomials,” Canadian Journal of Mathematics, Vol. 48, 1996, pp. 1180-1195. doi:10.4153/CJM-1996-062-4 [6] A. J. Duran and W. Van Assche, “Orthogonal Matrix Polynomials and Higher Order Recurrence Relations,” Linear Algebra and Its Applications, Vol. 219, No. 1, 1995, pp. 261-280. doi:10.1016/0024-3795(93)00218-O. [7] L. Jodar and J. Sastre, “The Growth of Laguerre Matrix Polynomials on Bounded Inervals,” Applied Mathematics Letters, Vol. 13, No. 8, 2000, pp. 21- 26. doi:10.1016/S0893-9659(00)00090-2. [8] L. Jodar, R. Company and E. Navarro, “Laguerre Matrix Polynomials and Systems of Second Order Differential Equations,” Applied Numerical Mathematics, Vol. 15, No. 1, 1994, pp. 53-63. doi:10.1016/0168-9274(94)00012-3. [9] A. Sinap and W. Van Assch, “Orthogonal Matrix Polynomials and Applications,” Journal of Computational and Applied Mathematics, Vol. 66, No. 1-2, 1996, pp. 27-52. doi:10.1016/0377-0427(95)00193-X. [10] S. Saks and A. Zygmund, “Analytic Functions,” Elsevier, Amsterdam, 1971. [11] N. Dunford and J. Schwartz, “Linear Operators, Part I,” Interscience, New York, 1955. [12] L. Jodar and J. C. Cortés, “On the Hypergeometric Matrix Function,” Journal of Computational and Applied Mathematics, Vol. 99, No. 1-2, 1998, pp. 205-217. doi:10.1016/S0377-0427(98)00158-7 [13] G. S. Khammash, “A Study of a Two Variables Gegenbauer Matrix Polynomials and Second Order Partial Matrix Differential Equations,” International Journal of Mathematical Analysis, Vol. 2, No. 17, 2008, pp. 807-821. [14] E. Defez and L. Jodar, “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. [15] E. Hille, “Lectures on Ordinary Differential Equations,” Addison-Wesley, New York, 1969. [16] L. Jodar and J. C. Cortés, “Some Properties of Gamma and Beta Matrix Functions,” Applied Mathematics Letters, Vol. 11, No. 1, 1998, pp. 89-93. doi:10.1016/S0893-9659(97)00139-0 [17] E. D. Rainvlle, “Special Functions,” Chelsea Publishing Co., Bronx, New York, 1960. [18] R. S. Batahan, “Anew Extension of Hermite Matrix Polynomials and Its Applications,” Linear Algebra and Its Applications, Vol. 419, No. 1, 2006, pp. 82-92. doi:10.1016/j.laa.2006.04.006
[1] A. G. Constantine and R. J. Mairhead, “Partial Differential Equations for Hypergeometric Functions of Two Argument Matrices,” Journal of Multivariate Analysis, Vol. 2, No. 3, 1972, pp. 332-338. doi:10.1016/0047-259X(72)90020-6. [2] R. J. Muirhead, “Systems of Partial Differential Equations for Hypergeometric Functions of Matrix Argument,” Annals of Mathematical Statistics, Vol. 40, No. 3, 1970, pp. 991-1001. doi:10.1214/aoms/1177696975 [3] A. Terras, “Special Functions the Symmetric Space of Positive Matrices,” SIAM Journal on Mathematical Analysis, Vol. 16, No. 3, 1985, pp. 620-640. doi:10.1137/0516046 [4] A. T. James, “Special Functions of Matrix and Single Argument in Statistics in Theory and Application of Special Functions,” Academic Press, New York, 1975. [5] A. J. Duran, “Markov’s Theorem for Orthogonal Matrix Polynomials,” Canadian Journal of Mathematics, Vol. 48, 1996, pp. 1180-1195. doi:10.4153/CJM-1996-062-4 [6] A. J. Duran and W. Van Assche, “Orthogonal Matrix Polynomials and Higher Order Recurrence Relations,” Linear Algebra and Its Applications, Vol. 219, No. 1, 1995, pp. 261-280. doi:10.1016/0024-3795(93)00218-O. [7] L. Jodar and J. Sastre, “The Growth of Laguerre Matrix Polynomials on Bounded Inervals,” Applied Mathematics Letters, Vol. 13, No. 8, 2000, pp. 21- 26. doi:10.1016/S0893-9659(00)00090-2. [8] L. Jodar, R. Company and E. Navarro, “Laguerre Matrix Polynomials and Systems of Second Order Differential Equations,” Applied Numerical Mathematics, Vol. 15, No. 1, 1994, pp. 53-63. doi:10.1016/0168-9274(94)00012-3. [9] A. Sinap and W. Van Assch, “Orthogonal Matrix Polynomials and Applications,” Journal of Computational and Applied Mathematics, Vol. 66, No. 1-2, 1996, pp. 27-52. doi:10.1016/0377-0427(95)00193-X. [10] S. Saks and A. Zygmund, “Analytic Functions,” Elsevier, Amsterdam, 1971. [11] N. Dunford and J. Schwartz, “Linear Operators, Part I,” Interscience, New York, 1955. [12] L. Jodar and J. C. Cortés, “On the Hypergeometric Matrix Function,” Journal of Computational and Applied Mathematics, Vol. 99, No. 1-2, 1998, pp. 205-217. doi:10.1016/S0377-0427(98)00158-7 [13] G. S. Khammash, “A Study of a Two Variables Gegenbauer Matrix Polynomials and Second Order Partial Matrix Differential Equations,” International Journal of Mathematical Analysis, Vol. 2, No. 17, 2008, pp. 807-821. [14] E. Defez and L. Jodar, “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. [15] E. Hille, “Lectures on Ordinary Differential Equations,” Addison-Wesley, New York, 1969. [16] L. Jodar and J. C. Cortés, “Some Properties of Gamma and Beta Matrix Functions,” Applied Mathematics Letters, Vol. 11, No. 1, 1998, pp. 89-93. doi:10.1016/S0893-9659(97)00139-0 [17] E. D. Rainvlle, “Special Functions,” Chelsea Publishing Co., Bronx, New York, 1960. [18] R. S. Batahan, “Anew Extension of Hermite Matrix Polynomials and Its Applications,” Linear Algebra and Its Applications, Vol. 419, No. 1, 2006, pp. 82-92. doi:10.1016/j.laa.2006.04.006