Generalized Form of Hermite Matrix Polynomials via the Hypergeometric Matrix Function
Author(s)
Raed S. Batahan*
Affiliation(s)
Department of Mathematics, Faculty of Science, Hadhramout University, Hadhranout, Yemen.
Department of Mathematics, Faculty of Science, Hadhramout University, Hadhranout, Yemen.
ABSTRACT
The object of this paper is to present a new generalization of the Hermite matrix polynomials by means of the hypergeometric matrix function. An integral representation, differential recurrence relation and some other properties of these generalized forms are established here. Moreover, some new properties of the Hermite and Chebyshev matrix polynomials are obtained. In particular, the two-variable and two-index Chebyshev matrix polynomials of two matrices are presented.
KEYWORDS
Hermite and Chebyshev Matrix Polynomials, Three Terms Recurrence Relation, Hypergeometric Matrix Function and Gamma Matrix Function
Hermite and Chebyshev Matrix Polynomials, Three Terms Recurrence Relation, Hypergeometric Matrix Function and Gamma Matrix Function
Cite this paper
Batahan, R. (2014) Generalized Form of Hermite Matrix Polynomials via the Hypergeometric Matrix Function. Advances in Linear Algebra & Matrix Theory, 4, 134-141. doi: 10.4236/alamt.2014.42012.
Batahan, R. (2014) Generalized Form of Hermite Matrix Polynomials via the Hypergeometric Matrix Function. Advances in Linear Algebra & Matrix Theory, 4, 134-141. doi: 10.4236/alamt.2014.42012.
References
[1] Jódar, L., Defez, E. and Ponsoda, E. (1996) Orthogonal Matrix Polynomials with Respect to Linear Matrix Moment Functionals: Theory and Applications. Approximation Theory and Its Applications, 12, 96-115. [2] Jódar, L. and Company, R. (1996) Hermite Matrix Polynomials and Second Order Matrix Differential Equations. Approximation Theory and Its Applications, 12, 20-30. [3] Jódar, L. and Defez, E. (1998) On Hermite Matrix Polynomials and Hermite Matrix Function. Approximation Theory and Its Applications, 14, 36-48. [4] Batahan, R.S. (2006) A New Extension of Hermite Matrix Polynomials and Its Applications. Linear Algebra and its Applications, 419, 82-92.
http://dx.doi.org/10.1016/j.laa.2006.04.006 [5] Batahan, R.S. and Bathanya, A.A. (2012) A New Generalization of Two-Variable Hermite Matrix Polynomials. Global Journal of Pure and Applied Mathematics, 8, 383-393. [6] Defez, E., Tung, M.M. and Sastre, J. (2011) Improvement on the Bound of Hermite Matrix Polynomials. Linear Algebra and its Applications, 434, 1910-1919.
http://dx.doi.org/10.1016/j.laa.2010.12.015 [7] Khan, S. and Raza, N. (2010) 2-Variable Generalized Hermite Matrix Polynomials and Lie Algebra Representation. Reports on Mathematical Physics, 66, 159-174.
http://dx.doi.org/10.1016/j.laa.2010.12.015 [8] Metwally, M.S. (2011) Operational Rules and Arbitrary Order Two-Index Two-Variable Hermite Matrix Generating Functions. Acta Mathematica Academiae Paedagogicae Nyegyhiensi (N.S.), 27, 41-49. [9] Metwally, M.S., Mohamed, M.T. and Shehata, A. (2009) Generalizations of Two-Index Two-Variable Hermite Matrix Polynomials. Demonstratio Mathematica, 42, 687-701. [10] Sayyed, K.A.M., Metwally, M.S. and Batahan, R.S. (2003) On Gegeralized Hermite Matrix Polynomials. Electronic Journal of Linear Algebra, 10, 272-279. [11] Shahwan, M.J.S. and Pathan, M.A. (2006) Origin of Certain Generating Relations of Hermite Matrix Functions from the View Point of Lie Algebra. Integral Transforms and Special Functions, 17, 734-747.
http://dx.doi.org/10.1080/10652460600725069 [12] Altin, A. and Çekim, B. (2012) Generating Matrix Functions for Chebyshev Matrix Polynomials of the Second Kind. Hacettepe Journal of Mathematics and Statistics, 41, 25-32. [13] Khan, S. and Al-Gonah, A.A. (2014) Multi-Variable Hermite Matrix Polynomials: Properties and Applications. Journal of Mathematical Analysis and Applications, 412, 222-235.
http://dx.doi.org/10.1016/j.jmaa.2013.10.037 [14] Dunford, N. and Schwartz, J. (1957) Linear Operators. 1, Interscience, New York. [15] Batahan, R.S. (2007) Generalized Gegenbauer Matrix Polynomials, Series Expansion and Some Properties. In: Ling, G.D., Ed., Linear Algebra Research Advances, Nova Science Publishers, 291-305. [16] Defez, E. and Jódar, L. (1998) Some Applications of the Hermite Matrix Polynomials Series Expansions. Journal of Computational and Applied Mathematics, 99, 105-117.
http://dx.doi.org/10.1016/S0377-0427(98)00149-6 [17] Hille, E. (1969) Lectures on Ordinary Differential Equations. Addison-Wesley, New York. [18] Jódar, L. and Cortés, J.C. (1998) Some Properties of Gamma and Beta Matrix Function. Applied Mathematics Letters, 11, 89-93.
http://dx.doi.org/10.1016/S0893-9659(97)00139-0 [19] Jódar, L. and Cortés, J.C. (1998) On the Hypergeometric Matrix Function. Journal of Computational and Applied Mathematics, 99, 205-217.
http://dx.doi.org/10.1016/S0377-0427(98)00158-7 [20] Defez, E. and Jódar, L. (2002) Chebyshev Matrix Polynomials and Second Order Matrix Differential Equations. Utilitas Mathematica, 61, 107-123.
[1] Jódar, L., Defez, E. and Ponsoda, E. (1996) Orthogonal Matrix Polynomials with Respect to Linear Matrix Moment Functionals: Theory and Applications. Approximation Theory and Its Applications, 12, 96-115. [2] Jódar, L. and Company, R. (1996) Hermite Matrix Polynomials and Second Order Matrix Differential Equations. Approximation Theory and Its Applications, 12, 20-30. [3] Jódar, L. and Defez, E. (1998) On Hermite Matrix Polynomials and Hermite Matrix Function. Approximation Theory and Its Applications, 14, 36-48. [4] Batahan, R.S. (2006) A New Extension of Hermite Matrix Polynomials and Its Applications. Linear Algebra and its Applications, 419, 82-92.
http://dx.doi.org/10.1016/j.laa.2006.04.006 [5] Batahan, R.S. and Bathanya, A.A. (2012) A New Generalization of Two-Variable Hermite Matrix Polynomials. Global Journal of Pure and Applied Mathematics, 8, 383-393. [6] Defez, E., Tung, M.M. and Sastre, J. (2011) Improvement on the Bound of Hermite Matrix Polynomials. Linear Algebra and its Applications, 434, 1910-1919.
http://dx.doi.org/10.1016/j.laa.2010.12.015 [7] Khan, S. and Raza, N. (2010) 2-Variable Generalized Hermite Matrix Polynomials and Lie Algebra Representation. Reports on Mathematical Physics, 66, 159-174.
http://dx.doi.org/10.1016/j.laa.2010.12.015 [8] Metwally, M.S. (2011) Operational Rules and Arbitrary Order Two-Index Two-Variable Hermite Matrix Generating Functions. Acta Mathematica Academiae Paedagogicae Nyegyhiensi (N.S.), 27, 41-49. [9] Metwally, M.S., Mohamed, M.T. and Shehata, A. (2009) Generalizations of Two-Index Two-Variable Hermite Matrix Polynomials. Demonstratio Mathematica, 42, 687-701. [10] Sayyed, K.A.M., Metwally, M.S. and Batahan, R.S. (2003) On Gegeralized Hermite Matrix Polynomials. Electronic Journal of Linear Algebra, 10, 272-279. [11] Shahwan, M.J.S. and Pathan, M.A. (2006) Origin of Certain Generating Relations of Hermite Matrix Functions from the View Point of Lie Algebra. Integral Transforms and Special Functions, 17, 734-747.
http://dx.doi.org/10.1080/10652460600725069 [12] Altin, A. and Çekim, B. (2012) Generating Matrix Functions for Chebyshev Matrix Polynomials of the Second Kind. Hacettepe Journal of Mathematics and Statistics, 41, 25-32. [13] Khan, S. and Al-Gonah, A.A. (2014) Multi-Variable Hermite Matrix Polynomials: Properties and Applications. Journal of Mathematical Analysis and Applications, 412, 222-235.
http://dx.doi.org/10.1016/j.jmaa.2013.10.037 [14] Dunford, N. and Schwartz, J. (1957) Linear Operators. 1, Interscience, New York. [15] Batahan, R.S. (2007) Generalized Gegenbauer Matrix Polynomials, Series Expansion and Some Properties. In: Ling, G.D., Ed., Linear Algebra Research Advances, Nova Science Publishers, 291-305. [16] Defez, E. and Jódar, L. (1998) Some Applications of the Hermite Matrix Polynomials Series Expansions. Journal of Computational and Applied Mathematics, 99, 105-117.
http://dx.doi.org/10.1016/S0377-0427(98)00149-6 [17] Hille, E. (1969) Lectures on Ordinary Differential Equations. Addison-Wesley, New York. [18] Jódar, L. and Cortés, J.C. (1998) Some Properties of Gamma and Beta Matrix Function. Applied Mathematics Letters, 11, 89-93.
http://dx.doi.org/10.1016/S0893-9659(97)00139-0 [19] Jódar, L. and Cortés, J.C. (1998) On the Hypergeometric Matrix Function. Journal of Computational and Applied Mathematics, 99, 205-217.
http://dx.doi.org/10.1016/S0377-0427(98)00158-7 [20] Defez, E. and Jódar, L. (2002) Chebyshev Matrix Polynomials and Second Order Matrix Differential Equations. Utilitas Mathematica, 61, 107-123.