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

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;

Hypergeometric Matrix Function;

Cite this paper

A. Shehata, "Certain*pl*(*m*,*n*)-Kummer Matrix Function of Two Complex Variables under Differential Operator," *Applied Mathematics*, Vol. 4 No. 1, 2013, pp. 91-96. doi: 10.4236/am.2013.41016.

A. Shehata, "Certain

References

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[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.

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[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.

[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.