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**Some relations satisfied by Hermite-Hermite matrix polynomials.**
*(English)*
Zbl 1424.33025

Summary: The classical Hermite-Hermite matrix polynomials for commutative matrices were first studied by M. S. Metwally et al. [Math. Bohem. 133, No. 4, 421–434 (2008; Zbl 1199.15079)]. 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.

### 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 |

33C50 | Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable |

33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |

44A45 | Classical operational calculus |