A New Extension of Humbert Matrix Function and Their Properties
Mohamed Abul-Dahab, Ayman Shehata
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DOI: 10.4236/apm.2011.16057   PDF    HTML     5,058 Downloads   9,969 Views   Citations

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.

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M. 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.

Conflicts of Interest

The authors declare no conflicts of interest.

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.

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