Certain pl(m,n)-Kummer Matrix Function of Two Complex Variables under Differential Operator

Ayman Shehata

Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt Department of Basic Applied Sciences, Unaizah Community College, Qassim University, Qassim, KSA.

DOI: 10.4236/am.2013.41016                

Abstract

The main aim of this paper is to define and study of a new matrix functions, say, the pl(m,n)-Kummer matrix function of two complex variables. The radius of regularity, recurrence relation and several new results on this function are established when the positive integers p is greater than one. Finally, we obtain a higher order partial differential equation satisfied by the pl(m,n)-Kummer matrix function and some special properties.

Keywords

Hypergeometric Matrix Function; pl(m, n)-Kummer Matrix Function; Matrix Differential Equation;

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. G. Constantine and R. J. Mairhead, “Partial Differential Equations for Hyper-Geometric 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]	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]	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
[4]	A. M. Mathai, “A Handbook of Generalized Special Functions for Statistical and Physical Sciences,” Oxford University Press, Oxford, 1993.
[5]	A. M. Mathai, “Jacobians of Matrix Transformations and Functions of Matrix Argument,” World Scientific Publishing, New York, 1997.
[6]	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
[7]	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. http://www.jams.or.jp/notice/scmjol/2009.html#2009-21
[8]	M. T. Mohamed and A. Shehata, “A Study of Appell’s Matrix Functions of Two Complex Variables and Some Properties,” Advances and Applications in Mathematical Sciences, Vol. 9, No. 1, 2011, pp. 23-33.
[9]	Z. M. G. Kishka, A. Shehata and M. Abul-Dahab, “A New Extension of Hypergeometric Matrix Functions,” Advances and Applications in Mathematical Sciences, Vol. 10, No. 4, 2011, pp. 349-371.
[10]	L. Jódar and J. C. Cortés, “Closed form General Solution of the Hypergeometric Matrix Differential Equation,” Mathematical and Computer Modelling, Vol. 32, No. 9, 2000, pp. 1017-1028. doi:10.1016/S0895-7177(00)00187-4
[11]	A. Shehata, “A Study of Some Special Functions and Polynomials of Complex Variables,” Ph.D. Thesis, Assiut University, Assiut, 2009.
[12]	A. Shehata, “On p- and q-Horn’s Matrix Function of Two Complex Variables,” Applied Mathematics, Vol. 2, No. 12, 2011, pp. 1437-1442. doi:10.4236/am.2011.212203
[13]	A. Shehata, “On Pseudo Legendre Matrix Polynomials,” International Journal of Mathematical Sciences and Engineering Applications (IJMSEA), Vol. 6, No. 6, 2012, pp. 251-258.
[14]	Z. M. G. Kishka, M. A. Saleem, S. Z. Radi and M. Abul- Dahab, “On the p- and q-Appell Matrix Function,” South-East Asian Bulletin of Mathematics, Vol. 35, 2011, pp. 807-818.
[15]	M. S. Metwally, “On p-Kummers Matrix Function of Complex Variable under Differential Operators and Their Properties,” South-East Asian Bulletin of Mathematics, Vol. 35, 2011, pp. 1-16.
[16]	A. Shehata and M. Abul-Dahab, “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
[17]	G. Golub and C. F. Van Loan, “Matrix Computations,” The Johns Hopkins University Press, Baltimore, 1989.
[18]	N. Dunford and J. Schwartz, “Linear Operators, Part I,” Interscience, New York, 1955.
[19]	K. A. M. Sayyed, “Basic Sets of Polynomials of Two Complex Variables and Convergence Properties,” Ph.D. Thesis, Assiut University, Assiut, 1975.