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.


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


Zbl 1199.15079
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