On a multivariable extension of Jacobi matrix polynomials. (English) Zbl 1221.33022

Summary: The classical Jacobi matrix polynomials only for commutative matrices were first studied by E. Defez, L. Jódar and A. Law [Comput. Math. Appl. 48, No. 5–6, 789–803 (2004; Zbl 1069.33007)]. The main aim of this paper is to construct a multivariable extension with the help of the classical Jacobi matrix polynomials (JMPs). Generating matrix functions and recurrence relations satisfied by these multivariable matrix polynomials are derived. Furthermore, general families of multilinear and multilateral generating matrix functions are obtained and their applications are presented.


33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
15B57 Hermitian, skew-Hermitian, and related matrices


Zbl 1069.33007
Full Text: DOI


[1] Constantine, A.G.; Muirhead, R.J., Partial differential equations for hypergeometric functions of two argument matrix, J. multivariate anal., 3, 332-338, (1972) · Zbl 0256.33007
[2] Terras, A., Special functions for the symmetric space of positive matrices, SIAM J. math. anal., 16, 620-640, (1985) · Zbl 0574.10030
[3] James, A.T., Special functions of matrix and single argument in statistics, (), 497-520
[4] A. Altın, B. Çekim, E. Erkuş-Duman, Families of generating functions for the Jacobi and related matrix polynomials, Ars Combin. (in press).
[5] Defez, E.; Jódar, L., Chebyshev matrix polynomials and second order matrix differential equations, Util. math., 61, 107-123, (2002) · Zbl 0998.15034
[6] Defez, E.; Jódar, L.; Law, A.; Ponsoda, E., Three-term recurrences and matrix orthogonal polynomials, Util. math., 57, 129-146, (2000) · Zbl 0962.05064
[7] 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
[8] Jódar, L.; Company, R.; Ponsoda, E., Orthogonal matrix polynomials and systems of second order differential equations, Differ. equ. dyn. syst., 3, 269-288, (1996) · Zbl 0892.33004
[9] Duran, A.J.; Van Assche, W., Orthogonal matrix polynomials and higher order recurrence relations, Linear algebra appl., 219, 261-280, (1995) · Zbl 0827.15027
[10] Duran, A.J.; López-Rodriguez, P., Orthogonal matrix polynomials: zeros and blumenthal’s theorem, J. approx. theory, 84, 96-118, (1996) · Zbl 0861.42016
[11] Jódar, L.; Defez, E.; Ponsoda, E., Matrix quadrature and orthogonal matrix polynomials, Congr. numer., 106, 141-153, (1995) · Zbl 0836.33004
[12] Jódar, L.; Sastre, J., The growth of Laguerre matrix polynomials on bounded intervals, Appl. math. lett., 13, 21-26, (2000) · Zbl 1112.33300
[13] Jódar, L.; Defez, E.; Ponsoda, E., Orthogonal matrix polynomials with respect to linear matrix moment functionals: theory and applications, J. approx. theory appl., 12, 1, 96-115, (1996) · Zbl 0853.42022
[14] Batahan, Raed S., A new extension of Hermite matrix polynomials and its applications, Linear algebra appl., 419, 82-92, (2006) · Zbl 1106.15016
[15] Defez, E.; Jódar, L., Some applications of the Hermite matrix polynomials series expansions, J. comput. appl. math., 99, 105-117, (1998) · Zbl 0929.33006
[16] Jódar, L.; Defez, E., A connection between laguerre’s and hermite’s matrix polynomials, Appl. math. lett., 11, 13-17, (1998) · Zbl 1074.33011
[17] Sayyed, K.A.M.; Metwally, M.S.; Batahan, R.S., On generalized Hermite matrix polynomials, Electron. J. linear algebra, 10, 272-279, (2003) · Zbl 1038.33005
[18] 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
[19] Jódar, L.; Cortés, J.C., On the hypergeometric matrix function, J. comput. appl. math., 99, 205-217, (1998) · Zbl 0933.33004
[20] Jódar, L.; Cortés, J.C., Closed form general solution of the hypergeometric matrix differential equation, Math. comput. modelling, 32, 1017-1028, (2000) · Zbl 0985.33006
[21] Sayyed, K.A.M.; Metwally, M.S.; Batahan, R.S., Gegenbauer matrix polynomials and second order matrix differential equations, Divulg. mat., 12, 101-115, (2004) · Zbl 1102.33010
[22] Dunford, N.; Schwartz, J., Linear operators, vol. I, (1957), Interscience New York
[23] Srivastava, H.M.; Manocha, H.L., A treatise on generating functions, (1984), Halsted Press, Ellis Horwood Limited, Chichester, John Wiley and Sons New York · Zbl 0535.33001
[24] Özarslan, M.A.; Altın, A., Some families of generating functions for the multiple orthogonal polynomials associated with modified Bessel \(K\)-functions, J. math. anal. appl., 297, 1, 186-193, (2004) · Zbl 1057.33006
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