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A New Extension of Generalized Hermite Matrix Polynomials

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Abstract

The Hermite matrix polynomials have been generalized in a number of ways, and many of these generalizations have been shown to be important tools in applications. In this paper, we introduce a new generalization of the Hermite matrix polynomials and present the recurrence relations and the expansion of these new generalized Hermite matrix polynomials. We also give new series expansions of the matrix functions \(\exp (xB)\), \(\sin (xB)\), \(\cos (xB)\), \(\cosh (xB)\) and \(\sinh (xB)\) in terms of these generalized Hermite matrix polynomials and thus prove that many of the seemingly different generalizations of the Hermite matrix polynomials may be viewed as particular cases of the two-variable polynomials introduced here. The generalized Chebyshev and Legendre matrix polynomials have also been introduced in this paper in terms of these generalized Hermite matrix polynomials.

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Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions which have led to the better presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Municipal Post Graduate College, Mussoorie, Dehradun, 248179, Uttarakhand, India

    Lalit Mohan Upadhyaya

  2. Department of Mathematics, College of Science and Arts in Unaizah, Qassim University, Qassim, 10363, Kingdom of Saudi Arabia

    Ayman Shehata

  3. Department of Mathematics, Faculty of Science, Assiut University, Assiut, 1516, Egypt

    Ayman Shehata

Authors
  1. Lalit Mohan Upadhyaya
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  2. Ayman Shehata
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Corresponding author

Correspondence to Ayman Shehata.

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Communicated by Ali Hassan Mohamed Murid.

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Upadhyaya, L.M., Shehata, A. A New Extension of Generalized Hermite Matrix Polynomials. Bull. Malays. Math. Sci. Soc. 38, 165–179 (2015). https://doi.org/10.1007/s40840-014-0010-3

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  • DOI: https://doi.org/10.1007/s40840-014-0010-3

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