Back
 ALAMT  Vol.8 No.2 , June 2018
On Horn Matrix Function H2(A,A′,B,B′;C;z,w) of Two Complex Variables under Differential Operator
Abstract: The aim of this paper deals with the study of the Horn matrix function of two complex variables. The convergent properties, an integral representation of H2(A,A′,B,B′;C;z,w) is obtained and recurrence matrix relations are given. Some result when operating on Horn matrix function with the differential operator D and a solution of certain partial differential equations are established. The Hadamard product of two Horn’s matrix functions is studied, certain results as, the domain of regularity, contiguous functional relations and operating with the differential operator D and D2 are established.
Cite this paper: Metwally, M. , Mohamed, M. , Shehata, A. (2018) On Horn Matrix Function H2(A,A′,B,B′;C;z,w) of Two Complex Variables under Differential Operator. Advances in Linear Algebra & Matrix Theory, 8, 96-110. doi: 10.4236/alamt.2018.82009.
References

[1]   Constantine, A.G. and Mairhead, R.J. (1972) Partial Differential Equations for Hypergeometric Functions of Two Argument Matrices. Journal of Multivariate Analysis, 2, 332-338.
https://doi.org/10.1016/0047-259X(72)90020-6

[2]   James, A.T. (1975) Special Functions of Matrix and Single Argument in Statistics in Theory and Application of Special Functions. In: Askey, R.A., Ed., Academic Press, New York.

[3]   Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1953) Higher Tanscendental Functions. McGraw-Hill Book Co., New York.

[4]   Mathai, A.M. (1993) A Handbook of Generalized Special Functions for Statistical and Physical Sciences. Oxford University Press, Oxford.

[5]   Mathai, A.M. (1997) Jacobians of Matrix Transformations and Functions of Matrix Argument. World Scientific Publishing, New York.
https://doi.org/10.1142/3438

[6]   Jódar, L. and Cortés, J.C. (1998) On the Hypergeometric Matrix Function. Journal of Computational and Applied Mathematics, 99, 205-217.
https://doi.org/10.1016/S0377-0427(98)00158-7

[7]   Jódar, L. and Cortés, J.C. (2000) Closed Form General Solution of the Hypergeometric Matrix Differential Equation. Mathematical and Computer Modelling, 32, 1017-1028.
https://doi.org/10.1016/S0895-7177(00)00187-4

[8]   Sayyed, K.A.M., Metwally, M.S. and Mohamed, M.T. (2009) Certain Hypergeometric Matrix Function. Scientiae Mathematicae Japonicae, 69, 315-321.
http://www.jams.or.jp/notice/scmjol/2009.html2009-21

[9]   Khammash, G.S. and Shehata, A. (2012) A Study on Some Prop-erties of Legendre Polynomials and Integral Transforms. International Journal of Open Problems in Computer Science and Mathe-matics (IJOPCM), 5, 1-6.
https://doi.org/10.12816/0006100

[10]   Khammash, G.S. and Shehata, A. (2012) On Humbert Matrix Polynomials. Asian Journal of Current Engineering and Maths (AJCEM), 1, 232-240.

[11]   Metwally, M.S. (2011) On p-Kummers Matrix Function of Complex Variable under Differential Operators and Their Properties. Southeast Asian Bulletin of Mathematics, 35, 1-16.

[12]   Metwally, M.S., Mohamed, M.T. and Shehata, A. (2008) On Hermite-Hermite Matrix Polynomials. Mathematica Bohemica, 133, 421-434.

[13]   Metwally, M.S. and Shehata, A. (2013) A Study of a New Properties of Hermite-Hermite Matrix Polynomials and Their Applications. International Journal of Mathematical Sciences and Engineering Applications (IJMSEA), 7, 19-33.

[14]   Rashwan, R.A., Metwally, M.S., Mohamed, M.T. and Shehata, A. (2013) Certain Kummer’s Matrix Function of Two Complex Variables under Certain Differential and Integral Operators. Thai Journal of Mathematics, 11, 725-740.

[15]   Shehata, A. (2012) A New Extension of Hermite-Hermite Matrix Polynomials and Their Properties. Thai Journal of Mathematics, 10, 433-444.

[16]   Shehata, A. (2012) A New Extension of Gegenbauer Matrix Polynomials and Their Properties. Bulletin of International Mathematical Virtual Institute, 2, 29-42.

[17]   Shehata, A. (2012) On Pseudo Legendre Matrix Polynomials. International Journal of Mathematical Sciences and Engineering Applications, 6, 251-258.

[18]   Upadhyaya, L.M. and Shehata, A. (2011) On Legendre Matrix Polynomials and Its Applications. International Transactions in Mathematical Sciences and Computer, 4, 291-310.

[19]   Shehata, A. (2011) On p and q-Horn’s Matrix Function of Two Complex Variables. Applied Mathematics, 2, 1437-1442.
https://doi.org/10.4236/am.2011.212203

[20]   Shehata, A. (2013) Certain pl(m,n)-Kummer Matrix Function of Two Complex Variables under Differential Operator. Applied Mathematics, 4, 91-96.
https://doi.org/10.4236/am.2013.41016

[21]   Horn, J. (1931) Hypergeometrische Funktionen zweier Veränderlichen. Mathematische Annalen, 105, 381-407.
https://doi.org/10.1007/BF01455825

[22]   Horn, J. (1935) Hypergeometrische Funktionen zweier Veränderlichen. Mathematische Annalen, 111, 638-677.
https://doi.org/10.1007/BF01472246

[23]   Horn, J. (1936) Hypergeometrische Funktionen zweier Veränderlichen. Mathematische Annalen, 113, 242-291.
https://doi.org/10.1007/BF01571633

[24]   Horn, J. (1938) Hypergeometrische Funktionen zweier Veränderlichen im Schnittpunkt dreier Singularitäten. Mathematische Annalen, 115, 435-455.
https://doi.org/10.1007/BF01448950

[25]   Jódar, L. and Cortés, J.C. (1998) Some Properties of Gamma and Beta Matrix Functions. Applied Mathematics Letters, 11, 89-93.
https://doi.org/10.1016/S0893-9659(97)00139-0

[26]   Dunford, N. and Schwartz, J.T. (1957) Linear Operators, Part I, General Theory. Interscience Publishers, Inc., New York.

[27]   Shehata, A. (2009) A Study of Some Special Functions and Polynomials of Complex Variables. PhD Thesis, Assiut University, Assiut.

[28]   Sayyed, K.A.M. (1975) Basic Sets of Polynomials of Two Complex Variables and Convergence Propertiess. PhD Thesis, Assiut University, Assiut.

 
 
Top