## A new extension of generalized Hermite matrix polynomials.(English)Zbl 1311.33008

Summary: The Hermite matrix polynomials have been generalized in a number of ways, and many of these generalizations have been shown to be important tools in applications. In this paper, we introduce a new generalization of the Hermite matrix polynomials and present the recurrence relations and the expansion of these new generalized Hermite matrix polynomials. We also give new series expansions of the matrix functions $$\exp (xB)$$, $$\sin (xB)$$, $$\cos (xB)$$, $$\cosh (xB)$$ and $$\sinh (xB)$$ in terms of these generalized Hermite matrix polynomials and thus prove that many of the seemingly different generalizations of the Hermite matrix polynomials may be viewed as particular cases of the two-variable polynomials introduced here. The generalized Chebyshev and Legendre matrix polynomials have also been introduced in this paper in terms of these generalized Hermite matrix polynomials.

### MSC:

 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 33E20 Other functions defined by series and integrals 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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### References:

 [1] Batahan, RS, A new extension of Hermite matrix polynomials and its applications, Linear. Algebra. Appl., 419, 82-92, (2006) · Zbl 1106.15016 [2] Defez, E; Jódar, L, Some applications of the Hermite matrix polynomials series expansions, J. Comput. Appl. Math., 99, 105-117, (1998) · Zbl 0929.33006 [3] Defez, E; Jódar, L, Chebyshev matrix polynomials and second order matrix differential equations, Util. Math., 61, 107-123, (2002) · Zbl 0998.15034 [4] Defez, E; Jódar, L; Law, A, Jacobi matrix differential equation, polynomial solutions, and their properties, Comput. Math. Appl., 48, 789-803, (2004) · Zbl 1069.33007 [5] Defez, E; Hervás, A; Jódar, L; Law, A, Bounding Hermite matrix polynomials, Math. Comput. Model., 40, 117-125, (2004) · Zbl 1061.33007 [6] Dunford, N., Schwartz, J.: Linear Operators, vol. I. Interscience, New York (1957) · Zbl 0635.47003 [7] Jódar, L; Company, R, Hermite matrix polynomials and second order matrix differential equations, J. Approx. Theory. Appl., 12, 20-30, (1996) · Zbl 0858.15014 [8] Jódar, L; Company, R; Navarro, E, Laguerre matrix polynomials and system of second-order differential equations, Appl. Numer. Math., 15, 53-63, (1994) · Zbl 0821.34010 [9] Jódar, L; Company, R; Ponsoda, E, Orthogonal matrix polynomials and systems of second order differential equations, Differ. Equ. Dyn. Syst., 3, 269-288, (1995) · Zbl 0892.33004 [10] Jódar, L; Cortés, JC, Closed form general solution of the hypergeometric matrix differential equation, Math. Comput. Model., 32, 1017-1028, (2000) · Zbl 0985.33006 [11] Jódar, L; Defez, E, A connection between laguerre’s and hermite’s matrix polynomials, Appl. Math. Lett., 11, 13-17, (1998) · Zbl 1074.33011 [12] Jódar, L; Defez, E, On Hermite matrix polynomials and Hermite matrix function, J. Approx. Theory. Appl., 14, 36-48, (1998) · Zbl 0911.15015 [13] Kahmmash, GS, A study of a two variables Gegenbauer matrix polynomials and second order matrix partial differential equations, Int. J. Math. Anal., 2, 807-821, (2008) · Zbl 1177.33011 [14] Khammash, GS, On Hermite matrix polynomials of two variables, J. Appl. Sci., 8, 1221-1227, (2008) [15] Metwally, MS; Mohamed, MT; Shehata, A, Generalizations of two-index two-variable Hermite matrix polynomials, Demonstr. Math., 42, 687-701, (2009) · Zbl 1186.15019 [16] Metwally, MS; Mohamed, MT; Shehata, A, On Hermite-Hermite matrix polynomials, Math. Bohem., 133, 421-434, (2008) · Zbl 1199.15079 [17] Rainville, E.D.: Special Functions. The Macmillan Company, New York (1960) · Zbl 0092.06503 [18] Sayyed, KAM; Metwally, MS; Batahan, RS, On generalized Hermite matrix polynomials, Electron. J. Linear. Algebra., 10, 272-279, (2003) · Zbl 1038.33005 [19] Sayyed, KAM; Metwally, MS; Batahan, RS, Gegenbauer matrix polynomials and second order matrix differential equations, Divulg. Mat., 12, 101-115, (2004) · Zbl 1102.33010 [20] Sinap, A; Assche, WV, Orthogonal matrix polynomials and applications, J. Comput. Appl. Math., 66, 27-52, (1996) · Zbl 0863.42018 [21] Srivastava, H.M., Karlsson, P.W.: Multiple Gaussian Hypergeometric Series. Wiley, New York (1985) · Zbl 0552.33001
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