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**Some relations on Humbert matrix polynomials.**
*(English)*
Zbl 1399.33016

Summary: The Humbert matrix polynomials were first studied by G. S. Khammash and A. Shehata [“On Humbert matrix polynomials”, Asian J. Current Eng. Maths. 1, No. 5, 232–240 (2012)]. Our goal is to derive some of their basic relations involving the Humbert matrix polynomials and then study several generating matrix functions, hypergeometric matrix representations, matrix differential equation and expansions in series of some relatively more familiar matrix polynomials of Legendre, Gegenbauer, Hermite, Laguerre and modified Laguerre. Finally, some definitions of generalized Humbert matrix polynomials also of two, three and several index are derived.

### MSC:

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |

33C55 | Spherical harmonics |

33E20 | Other functions defined by series and integrals |