A New Extension of Humbert Matrix Function and Their Properties

ABSTRACT

This paper deals with the study of the composite Humbert matrix function with matrix arguments . The convergence and integral form this function is established. An operational relation between a Humbert matrix function and Kummer matrix function is studied. Also, integral expressions of this relation are deduced. Finally, we define and study of the composite Humbert Kummer matrix functions.

This paper deals with the study of the composite Humbert matrix function with matrix arguments . The convergence and integral form this function is established. An operational relation between a Humbert matrix function and Kummer matrix function is studied. Also, integral expressions of this relation are deduced. Finally, we define and study of the composite Humbert Kummer matrix functions.

KEYWORDS

Hypergeometric Matrix function, Humbert Matrix Function, Kummer Matrix Function, Integral Representations

Hypergeometric Matrix function, Humbert Matrix Function, Kummer Matrix Function, Integral Representations

Cite this paper

nullM. Abul-Dahab and A. Shehata, "A New Extension of Humbert Matrix Function and Their Properties,"*Advances in Pure Mathematics*, Vol. 1 No. 6, 2011, pp. 315-321. doi: 10.4236/apm.2011.16057.

nullM. Abul-Dahab and A. Shehata, "A New Extension of Humbert Matrix Function and Their Properties,"

References

[1] A. G. Constantine and R. J. Mairhead, “Partial Differential Equations for Hypergeometric Function of Two Argument Matrix,” Journal of Multivariate Analysis, Vol. 2, No. 3, 1972, pp. 332-338. doi:10.1016/0047-259X(72)90020-6

[2] A.T. James, “Special Functions of Matrix and Single Argument in Statistics in Theory and Application of Special Functions,” Academic Press, New York, 1975.

[3] A. M. Mathai, “A Handbook of Generalized Special Functions for Statistical and Physical Sciences,” Oxford University Press, Oxford, 1993.

[4] A. M. Mathai, “Jacobians of Matrix Transformations and Functions of Matrix Argument,” World Scientific Publishing, New York, 1997.

[5] L. Jodar and E. Defez, “A Connection between Lagurre’s and Hermite’s Matrix Polynomials,” Applied Mathematics Letters, Vol. 11, 1998, pp. 13-17.

[6] E. Defez and L. Jódar, “Chebyshev Matrix Polynomails and Second Order Matrix Differential Equations,” Utilitas Mathematics, Vol. 61, 2002, pp. 107-123.

[7] 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

[8] J. Sastre and L. Jódar, “Asymptotics of the Modified Bessel and Incomplete Gamma Matrix Functions,” Applied Mathematics Letters, Vol. 16, No. 6, 2003, pp. 815- 820. doi:10.1016/S0893-9659(03)90001-2

[9] L. Jódar, R. Company and E. Navarro, “Bessel Matrix Functions: Explict Solution of Coupled Bessel Type Equations,” Utilitas Mathematics, Vol. 46, 1994, pp. 129-141.

[10] Z. M. G. Kishka, A. Shehata and M. Abul-Dahab, “On Humbert Matrix Function,” Applied Mathematics Letters, Article in Press.

[11] S. Z. Rida, M. Abul-Dahab, M. A. Saleem and M. T. Mohammed, “On Humbert Matrix Function Ψ1(A,B;C,C';z,w) of Two Complex Variables under Differential Operator,” International Journal of Industrial Mathematics, Vol. 32, 2010, pp. 167-179.

[12] N. N. Lebedev, “Special Functions and Their Applications,” Dover Publications Inc., New York, 1972.

[13] L.Y. Luke, “The Special Functions and Their Approximations,” Vol. 2, Academic Press, New York, 1969.

[14] H. M. Srivastava and P. W. Karlsson, “Multiple Gaussian Hypergeometric Series,” Ellis Horwood, Chichester, 1985.

[15] M. S. Metwally, “On p-Kummers Matrix Function of Complex Variable under Differential Operators and Their Properties,” Southeast Asian Bulletin of Mathematics, Vol. 35, 2011, pp. 1-16.

[16] A. Shehata, “A Study of Some Special Functions and Polynomials of Complex Variables,” Ph.D. Thesis, Assiut University, Assiut, 2009.

[17] L. Jódar 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

[18] G. Golub and C. F. Van Loan, “Matrix Computations,” The Johns Hopkins University Press, Baltimore, 1989.

[19] L. Jódar 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

[20] K. A. M. Sayyed, M. S. Metwally and M. T. Mohamed, “Certain Hypergeometric Matrix Function, ”Scientiae Mathematicae Japonicae, Vol. 69, No. 3, 2009, pp. 315- 321.

[1] A. G. Constantine and R. J. Mairhead, “Partial Differential Equations for Hypergeometric Function of Two Argument Matrix,” Journal of Multivariate Analysis, Vol. 2, No. 3, 1972, pp. 332-338. doi:10.1016/0047-259X(72)90020-6

[2] A.T. James, “Special Functions of Matrix and Single Argument in Statistics in Theory and Application of Special Functions,” Academic Press, New York, 1975.

[3] A. M. Mathai, “A Handbook of Generalized Special Functions for Statistical and Physical Sciences,” Oxford University Press, Oxford, 1993.

[4] A. M. Mathai, “Jacobians of Matrix Transformations and Functions of Matrix Argument,” World Scientific Publishing, New York, 1997.

[5] L. Jodar and E. Defez, “A Connection between Lagurre’s and Hermite’s Matrix Polynomials,” Applied Mathematics Letters, Vol. 11, 1998, pp. 13-17.

[6] E. Defez and L. Jódar, “Chebyshev Matrix Polynomails and Second Order Matrix Differential Equations,” Utilitas Mathematics, Vol. 61, 2002, pp. 107-123.

[7] 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

[8] J. Sastre and L. Jódar, “Asymptotics of the Modified Bessel and Incomplete Gamma Matrix Functions,” Applied Mathematics Letters, Vol. 16, No. 6, 2003, pp. 815- 820. doi:10.1016/S0893-9659(03)90001-2

[9] L. Jódar, R. Company and E. Navarro, “Bessel Matrix Functions: Explict Solution of Coupled Bessel Type Equations,” Utilitas Mathematics, Vol. 46, 1994, pp. 129-141.

[10] Z. M. G. Kishka, A. Shehata and M. Abul-Dahab, “On Humbert Matrix Function,” Applied Mathematics Letters, Article in Press.

[11] S. Z. Rida, M. Abul-Dahab, M. A. Saleem and M. T. Mohammed, “On Humbert Matrix Function Ψ1(A,B;C,C';z,w) of Two Complex Variables under Differential Operator,” International Journal of Industrial Mathematics, Vol. 32, 2010, pp. 167-179.

[12] N. N. Lebedev, “Special Functions and Their Applications,” Dover Publications Inc., New York, 1972.

[13] L.Y. Luke, “The Special Functions and Their Approximations,” Vol. 2, Academic Press, New York, 1969.

[14] H. M. Srivastava and P. W. Karlsson, “Multiple Gaussian Hypergeometric Series,” Ellis Horwood, Chichester, 1985.

[15] M. S. Metwally, “On p-Kummers Matrix Function of Complex Variable under Differential Operators and Their Properties,” Southeast Asian Bulletin of Mathematics, Vol. 35, 2011, pp. 1-16.

[16] A. Shehata, “A Study of Some Special Functions and Polynomials of Complex Variables,” Ph.D. Thesis, Assiut University, Assiut, 2009.

[17] L. Jódar 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

[18] G. Golub and C. F. Van Loan, “Matrix Computations,” The Johns Hopkins University Press, Baltimore, 1989.

[19] L. Jódar 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

[20] K. A. M. Sayyed, M. S. Metwally and M. T. Mohamed, “Certain Hypergeometric Matrix Function, ”Scientiae Mathematicae Japonicae, Vol. 69, No. 3, 2009, pp. 315- 321.