Some relations satisfied by Hermite-Hermite matrix polynomials

Shehata, Ayman ; Upadhyaya, Lalit Mohan

Mathematica Bohemica, Tome 142 (2017) no. 2, pp. 145-162

Abstract (VO)

The classical Hermite-Hermite matrix polynomials for commutative matrices were first studied by Metwally et al. (2008). Our goal is to derive their basic properties including the orthogonality properties and Rodrigues formula. Furthermore, we define a new polynomial associated with the Hermite-Hermite matrix polynomials and establish the matrix differential equation associated with these polynomials. We give the addition theorems, multiplication theorems and summation formula for the Hermite-Hermite matrix polynomials. Finally, we establish general families and several new results concerning generalized Hermite-Hermite matrix polynomials.

Détail

DOI : 10.21136/MB.2016.0001-15

Classification : 15A60, 33C45, 33C50, 33C80, 34A25, 44A45
Keywords: Hermite-Hermite polynomials; matrix generating functions; orthogonality property; Rodrigues formula; associated Hermite-Hermite polynomials; generalized Hermite-Hermite matrix polynomials

Bibliographie

[1] Defez, E., Jódar, L.: Chebyshev matrix polynomials and second order matrix differential equations. Util. Math. 61 (2002), 107-123. | MR | JFM

[2] Defez, E., Jódar, L., Law, A.: Jacobi matrix differential equation, polynomial solutions, and their properties. Comput. Math. Appl. 48 (2004), 789-803. | DOI | MR | JFM

[3] Defez, E., Jódar, L., Law, A., Ponsoda, E.: Three-term recurrences and matrix orthogonal polynomials. Util. Math. 57 (2000), 129-146. | MR | JFM

[4] Dunford, N., Schwartz, J. T.: Linear Operators. I. General Theory. Pure and Applied Mathematics 7. Interscience Publishers, New York (1958). | MR | JFM

[5] Durán, A. J., Assche, W. Van: Orthogonal matrix polynomials and higher-order recurrence relations. Linear Algebra Appl. 219 (1995), 261-280. | DOI | MR | JFM

[6] Jódar, L., Company, R.: Hermite matrix polynomials and second order matrix differential equations. Approximation Theory Appl. 12 (1996), 20-30. | MR | JFM

[7] Jódar, L., Company, R., Navarro, E.: Laguerre matrix polynomials and system of second-order differential equations. Appl. Numer. Math. 15 (1994), 53-63. | DOI | MR | JFM

[8] Jódar, L., Company, R., Ponsoda, E.: Orthogonal matrix polynomials and systems of second order differential equations. Differ. Equ. Dyn. Syst. 3 (1995), 269-288. | MR | JFM

[9] Jódar, L., Defez, E.: On Hermite matrix polynomials and Hermite matrix function. Approximation Theory Appl. 14 (1998), 36-48. | MR | JFM

[10] Khammash, G. S., Shehata, A.: On Humbert matrix polynomials. Asian J. Current Engineering and Maths. (AJCEM) 1 (2012), 232-240.

[11] Metwally, M. S., Mohamed, M. T., Shehata, A.: On Hermite-Hermite matrix polynomials. Math. Bohem. 133 (2008), 421-434. | MR | JFM

[12] Metwally, M. S., Mohamed, M. T., Shehata, A.: On pseudo Hermite matrix polynomials of two variables. Banach J. Math. Anal. (eletronic only) 4 (2010), 169-178. | DOI | MR | JFM

[13] Rainville, E. D.: Special Functions. Macmillan, New York (1960). | MR | JFM

[14] Sayyed, K. A. M., Metwally, M. S., Batahan, R. S.: On generalized Hermite matrix polynomials. Electron. J. Linear Algebra (eletronic only) 10 (2003), 272-279. | MR | JFM

[15] Shehata, A.: On $p$ and $q$-Horn's matrix function of two complex variables. Appl. Math., Irvine 2 (2011), 1437-1442. | DOI | MR

[16] Shehata, A.: On Tricomi and Hermite-Tricomi matrix functions of complex variable. Commun. Math. Appl. 2 (2011), 97-109. | MR | JFM

[17] Shehata, A.: A new extension of Gegenbauer matrix polynomials and their properties. Bull. Int. Math. Virtual Inst. 2 (2012), 29-42. | MR

[18] Shehata, A.: A new extension of Hermite-Hermite matrix polynomials and their properties. Thai J. Math. 10 (2012), 433-444. | MR | JFM

[19] Shehata, A.: On pseudo Legendre matrix polynomials. Int. J. Math. Sci. Eng. Appl. 6 (2012), 251-258. | MR

[20] Shehata, A.: Certain $pl(m,n)$-Kummer matrix function of two complex variables under differential operator. Appl. Math., Irvine 4 (2013), 91-96. | DOI

[21] Shehata, A.: On Rainville's matrix polynomials. Sylwan Journal 158 (2014), 158-178. | MR

[22] Shehata, A.: On Rice's matrix polynomials. Afr. Mat. 25 (2014), 757-777. | DOI | MR | JFM

[23] Shehata, A.: Connections between Legendre with Hermite and Laguerre matrix polynomials. Gazi University Journal of Science 28 (2015), 221-230. Avaible at http://gujs.gazi.edu.tr/article/view/1060001688/5000116963

[24] Shehata, A.: New kinds of hypergeometric matrix functions. British Journal of Mathematics and Computer Science. 5 (2015), 92-103. | DOI | MR

[25] Shehata, A.: On a new family of the extended generalized Bessel-type matrix polynomials. Mitteilungen Klosterneuburg J. 65 (2015), 100-121.

[26] Shehata, A.: Some relations on Gegenbauer matrix polynomials. Review of Computer Engineering Research 2 (2015), 1-21. Avaible at http://www.pakinsight.com/pdf-files/RCER-2015-2\%281\%29-1-21.pdf | DOI

[27] Shehata, A.: Some relations on Humbert matrix polynomials. Math. Bohem. 141 (2016), 407-429. | DOI | MR | JFM

[28] Shehata, A.: Some relations on Konhauser matrix polynomials. Miskolc Math. Notes 17 (2016), 605-633. | DOI | MR

[29] Sinap, A., Assche, W. Van: Orthogonal matrix polynomials and applications. Proc. 6th Int. Congress on Computational and Applied Mathematics (F. Broeckx at al., eds). Leuven, 1994 J. Comput. Appl. Math. 66, Elsevier Science, Amsterdam (1996), 27-52. | DOI | MR | JFM

[30] Upadhyaya, L. M., Shehata, A.: On Legendre matrix polynomials and its applications. Inter. Tran. Math. Sci. Comput. 4 (2011), 291-310. | MR

[31] Upadhyaya, L. M., Shehata, A.: A new extension of generalized Hermite matrix polynomials. Bull. Malays. Math. Sci. Soc. (2) 38 (2015), 165-179. | DOI | MR | JFM