On *p* and *q*-Horn’s Matrix Function of Two Complex Variables

Author(s)
Ayman Shehata

ABSTRACT

The main aim of this paper is to define and study of a new Horn’s matrix function, say, the p and q-Horn’s matrix function of two complex variables. The radius of regularity on this function is given when the positive integers p and q are greater than one, an integral representation of^{p}H^{q} _{2} is obtained, recurrence relations are established. Finally, we obtain a higher order partial differential equation satisfied by the *p* and *q*-Horn’s matrix function.

The main aim of this paper is to define and study of a new Horn’s matrix function, say, the p and q-Horn’s matrix function of two complex variables. The radius of regularity on this function is given when the positive integers p and q are greater than one, an integral representation of

KEYWORDS

Hypergeometric Matrix functions, p and q-Horn’s Matrix Function, Contiguous Relations, Matrix Functions, Matrix Differential Equation, Differential Operator

Hypergeometric Matrix functions, p and q-Horn’s Matrix Function, Contiguous Relations, Matrix Functions, Matrix Differential Equation, Differential Operator

Cite this paper

nullA. 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.

nullA. Shehata, "On

References

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

[2] 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

[3] A. M. Mathai, “A Handbook of Generalized Spcial 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] H. M. Srivastava and H. L. Manocha, “A Treatise on Generating Functions,” Ellis Horwood, New York, 1984.

[6] M. T. Mohamed and A. Shehata, “A Study of Appell’s Matrix Functions of Two Complex Variables and Some Properties,” Journal Advances and Applications in Mathematical Sciences, Vol. 9, No. 1, 2011, pp. 23-33.

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

[8] 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

[9] 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

[10] L. M. Upadhyaya and H. S. Dhami, “Generalized Horn’s Functions of Matrix Arguments,” Bulletin Pure and Applied Sciences: Section E. Mathematics and Statistics, Vol. 29 E, No. 2, 2010, pp. 353–364.

[11] L. M. Upadhyaya, “A Summation Formula for a Horn’s Double Hypergeometric Function-II,” Bulletin Pure and Applied Sciences: Section E. Mathematics and Statistics, Vol. 2 E, No. 2, 2010, pp. 279-286.

[12] Z. M. G. Kishka, M. A. Saleem, S. Z. Radi and M. Abul-Dahab, “On the p and q-Appell Matrix Function,” Southeast Asian Bulletin of Mathematics, Vol. 35, 2011, pp. 807-818.

[13] 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.

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

[15] N. Dunford and J. Schwartz, “Linear Operators, Part I,” Interscience, New York, 1955.

[16] 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

[17] K. A. M. Sayyed, “Basic Sets of Polynomials of Two Complex Variables and Convergence Properties,” Ph.D. Thesis, Assiut University, Assiut, Egypt, 1975.

[18] 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

[19] A. Erdélyi, W. Magnus, F. Oberhettinger and G. Tricomi, “Higher Tanscendental Functions,” McGraw-Hill Book Co., New York, Vol. 1, 1953.

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

[2] 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

[3] A. M. Mathai, “A Handbook of Generalized Spcial 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] H. M. Srivastava and H. L. Manocha, “A Treatise on Generating Functions,” Ellis Horwood, New York, 1984.

[6] M. T. Mohamed and A. Shehata, “A Study of Appell’s Matrix Functions of Two Complex Variables and Some Properties,” Journal Advances and Applications in Mathematical Sciences, Vol. 9, No. 1, 2011, pp. 23-33.

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

[8] 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

[9] 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

[10] L. M. Upadhyaya and H. S. Dhami, “Generalized Horn’s Functions of Matrix Arguments,” Bulletin Pure and Applied Sciences: Section E. Mathematics and Statistics, Vol. 29 E, No. 2, 2010, pp. 353–364.

[11] L. M. Upadhyaya, “A Summation Formula for a Horn’s Double Hypergeometric Function-II,” Bulletin Pure and Applied Sciences: Section E. Mathematics and Statistics, Vol. 2 E, No. 2, 2010, pp. 279-286.

[12] Z. M. G. Kishka, M. A. Saleem, S. Z. Radi and M. Abul-Dahab, “On the p and q-Appell Matrix Function,” Southeast Asian Bulletin of Mathematics, Vol. 35, 2011, pp. 807-818.

[13] 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.

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

[15] N. Dunford and J. Schwartz, “Linear Operators, Part I,” Interscience, New York, 1955.

[16] 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

[17] K. A. M. Sayyed, “Basic Sets of Polynomials of Two Complex Variables and Convergence Properties,” Ph.D. Thesis, Assiut University, Assiut, Egypt, 1975.

[18] 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

[19] A. Erdélyi, W. Magnus, F. Oberhettinger and G. Tricomi, “Higher Tanscendental Functions,” McGraw-Hill Book Co., New York, Vol. 1, 1953.