Mathematica Bohemica, Vol. 142, No. 2, pp. 145-162, 2017

# Some relations satisfied by Hermite-Hermite matrix polynomials

## Ayman Shehata, Lalit Mohan Upadhyaya

#### Received January 5, 2015.   First published December 1, 2016.

Ayman Shehata (corresponding author), Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt, and Department of Mathematics, College of Science and Arts, Unaizah 51911, Qassim University, Qassim, Kingdom of Saudi Arabia, e-mail: drshehata2006@yahoo.com; Lalit Mohan Upadhyaya, Department of Mathematics, Municipal Post Graduate College, Mussoorie, Dehradun, Uttarakhand, India 248179, e-mail: lmupadhyaya@rediffmail.com

Abstract: The classical Hermite-Hermite matrix polynomials for commutative matrices were first studied by Metwally et al. (2008). 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.

Keywords: Hermite-Hermite polynomials; matrix generating functions; orthogonality property; Rodrigues formula; associated Hermite-Hermite polynomials; generalized Hermite-Hermite matrix polynomials

Classification (MSC 2010): 33C45, 34A25, 15A60, 44A45, 33C50, 33C80

DOI: 10.21136/MB.2016.0001-15

Full text available as PDF.

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