Skip to main content
SpringerLink
Log in

On the Discrete q-Hermite Matrix Polynomials

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

There are two definitions for the discrete q-Hermite polynomials, one of them is defined for \(0<q<1\) and the other is considered a generalization for \(q>1\). This paper is devoted to extend these definitions to the discrete q-Hermite matrix polynomials by means of the generating matrix functions. Explicit expressions and Rodrigues-type formulas for the discrete q-Hermite matrix polynomials are obtained. Some recurrence relations for these matrix polynomials, in particular the three terms recurrence relations are given. Furthermore, some identities are proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Jódar, L., Company, R., Navarro, E.: Laguerre matrix polynomials and system of second-order differential equations. Appl. Numer. Math. 15, 53–63 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Jódar, L., Company, R.: Hermite matrix polynomials and second order matrix differential equations. J. Approx. Theory Appl. 12(2), 20–30 (1996)

    MATH  MathSciNet  Google Scholar 

  3. Sayyed, K.A.M., Metwally, M.S., Batahan, R.S.: Gegenbauer matrix polynomials and second order matrix differential equations. Divulg. Mat. 12(2), 101–115 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Defez, E., Jódar, L.: Some applications of the Hermite matrix polynomials series expansions. J. Comput. Appl. Math. 99, 105–117 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Defez, E., Garcia-Honrubia, M., Villanueva, R.J.: Aprocedure for computing the exponential of a matrix using Hermite matrix polynomials. Far East J. Appl. Math. 6(3), 217–231 (2002)

    MATH  MathSciNet  Google Scholar 

  6. Jódar, L., Defez, E.: Some new matrix formulas related to Hermite matrix polynomials theory. In: M. Alfaro et al. (eds.) Proceedings of the International Workshop on Orthogonal Polynomials in Mathematical Physics, Legans, pp. 24–26 (1996)

  7. Jódar, L., Defez, E.: On Hermite matrix polynomials and Hermite matrix function. J. Approx. Theory Appl. 14(1), 36–48 (1998)

    MATH  MathSciNet  Google Scholar 

  8. Sayyed, K.A.M., Metwally, M.S., Batahan, R.S.: On generalized Hermite matrix polynomials. Electron. J. Linear Algebra 10, 272–279 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Batahan, R.S.: A new extension of Hermite matrix polynomials and its applications. Linear Algebra Appl. 419, 82–92 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dunford, N., Schwartz, J.: Linear Operators, Part I. Interscience, New York (1956)

    Google Scholar 

  11. Defez, E., Hervás, A., Jódar, L., Law, A.: Bounding Hermite matrix polynomials. Math. Comput. Model. 40, 117–125 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Andrews, G., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  13. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  14. Szego, G.: Beitrag zur Theorie der Thetafunktionen, Sitz. Preuss. Akad. Wiss. Phys. Math. Kl., XIX (1926) 242–252, Reprinted in “Collected Papers”, edited by R. Askey, vol. I, Birkhauser, Boston (1982)

  15. Koekoek, R., Swarttouw, R.F.: The Askeyscheme of hypergeometric orthogonal polynomials and its \(q\)-analogue. In: Report 98-17. Delft University of Technology, Delft (1998)

  16. Arik, M., Atakishiyev, N.M., Rueda, J.P.: Discrete \(q\)-Hermite polynomials are linked by the integral and finite Fourier transforms. Int. J. Differ. Equ. 1(2), 195–204 (2006)

    MATH  MathSciNet  Google Scholar 

  17. Salem, A.: On a \(q\)-gamma and a \(q\)-beta matrix functions. Linear Multilinear Algebra 60(6), 683–696 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Salem, A.: The basic Gauss hypergeometric matrix function and its matrix \(q\)-difference equation. Linear Multilinear Algebra 62(3), 347–361 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  19. Salem, A.: The q-Laguerre matrix polynomials. SpringerPlus 5, 550 (2016)

    Article  Google Scholar 

  20. Jackson, F.H.: \(q\)-form of Taylors theorem. Messenger Math. 39, 62–64 (1906)

    Google Scholar 

Download references

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for technical and financial support.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah, 21589, Saudi Arabia

    Ahmed Salem

Authors
  1. Ahmed Salem
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed Salem.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Salem, A. On the Discrete q-Hermite Matrix Polynomials. Int. J. Appl. Comput. Math 3, 3147–3158 (2017). https://doi.org/10.1007/s40819-016-0285-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40819-016-0285-1

Keywords

Mathematics Subject Classification

Search

Navigation