Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZATION OF MULTI-VARIABLE MODIFIED HERMITE MATRIX POLYNOMIALS AND ITS APPLICATIONS

  • Singh, Virender (Department of Applied Science, Mathematics, Galgotia college of Engineering and Technology) ;
  • Khan, Mumtaz Ahmad (Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University) ;
  • Khan, Abdul Hakim (Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University)
  • Received : 2019.05.05
  • Accepted : 2020.02.18
  • Published : 2020.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we get acquainted to a new generalization of the modified Hermite matrix polynomials. An explicit representation and expansion of the Matrix exponential in a series of these matrix polynomials is obtained. Some important properties of Modified Hermite Matrix polynomials such as generating functions, recurrence relations which allow us a mathematical operations. Also we drive expansion formulae and some operational representations.

Keywords

References

  1. P. Appell and J. Kampe de Feriet, Functions hyperge'ometriques et hyper-sphe'riques, polynome d' Hermite, Gauthier-villars, paris, 1926.
  2. R.S. Batahan, A new extension of Hermite matrix polynomials and its applications, Linear Algebra Appl. 419 (2006), 82-92. https://doi.org/10.1016/j.laa.2006.04.006
  3. R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, 1965.
  4. F. Brafman, Some generating functions for Laguerre and Hermite polynomials, Canadian Journal of Mathematics. 9 (1957), 180-187. https://doi.org/10.4153/CJM-1957-020-1
  5. G. Dattoli, S. Lorenzutta, G. Maino, A. Torre and C. Cesarano, Generalized Hermite polynomials and super-Gaussian forms, J. Math. Anal. Appl., 203 (3) (1996), 597-609. https://doi.org/10.1006/jmaa.1996.0399
  6. G. Dattoli, P.E. Ricci and I. Khomasuridze, On the derivation of new families of generating functions involving ordinary Bessel functions and Bessel-Hermite functions, Math. Comput. Modelling. 46 (3-4) (2007), 410-414. https://doi.org/10.1016/j.mcm.2006.11.011
  7. E. Defez and L. Jodar, Some applications of the Hermite matrix polynomials series expansions, J. Comp. Appl. Math. 99 (1998), 105-117. https://doi.org/10.1016/S0377-0427(98)00149-6
  8. N. Dunford and J. Schwartz, Linear Operators, part I, General Theory, Interscience Publishers, INC. New York, 1957.
  9. H.W. Gould and A.T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J., 29 (1962), 51-63. https://doi.org/10.1215/S0012-7094-62-02907-1
  10. L. Jodar and R. Company, Hermite Matrix Polynomials and Second Order Matrix Differential Equations, J. Approx. Theory Appl., 12 (2) (1996), 20-30.
  11. L. Jodar and J.C. Cortes, On the hypergeometric matrix function, J. Comp. Appl. Math., 99 (1998), 205-217. https://doi.org/10.1016/S0377-0427(98)00158-7
  12. L. Jodar and E. Defez, Some new matrix formulas related to Hermite matrix polynomials theory, Proceedings Int.Workshop Orth. Poly.Math. Phys. Leganes., (1996), 93-101.
  13. L. Jodar and E. Defez, A connection between Laguerre's and Hermite's matrix polynomials, Appl. Math. Lett., 11 (1998), 13-17.
  14. L. Jodar and E. Defez, On Hermite matrix polynomials and Hermite matrix function, J. Approx. Theory Appl., 14 (1998), 36-48.
  15. L. Jodar, R. Company and E. Navarro, Laguerre Matrix Polynomials and Systems of Second Order Differential Equations, Appl. Numer. Maths., 15 (1994), 53-63. https://doi.org/10.1016/0168-9274(94)00012-3
  16. L. Jodar, E. Defez and E. Ponsoda, Orthogonal matrix polynomials with respect to linear matrix moment functionals, Theory and applications, J. Approx. Theory Appl, 12 (1) (1996), 96-115.
  17. M. A. Khan and S. Ahmed, On legendre's polynomials of two variables, International Transactions in Applied sciences, 2 (1) (2002), 135-145.
  18. S. Khan and A.A. Al-Gonah, Multi-variable Hermite matrix polynomials properties and applications, J. Math. Anal. Appl. 412 (2014), 222-235. https://doi.org/10.1016/j.jmaa.2013.10.037
  19. S. Khan and N. Raza, 2-variable generalized Hermite matrix polynomials and lie algebra representation, Reports on mathematical physics, 66 (2) (2010), 159-174. https://doi.org/10.1016/S0034-4877(10)00024-8
  20. M.S. Metwally, M.T. Mohamed and A. Shehata, On Hermite-Hermite matrix polynomials, Math. Bohemica, 133 (2008), 421-434. https://doi.org/10.21136/MB.2008.140630
  21. M.S. Metwally, M.T. Mohamed and A. Shehata, Generalizations of two index two variable Hermite matrix polynomials, Demonstration Math., 42 (4) (2009), 687-701.
  22. I. Najfeld and T. F. Havel, Derivaties of the Matrix Exponential and their Computation, Advances in Appl. Maths., 16 (1995), 321-375. https://doi.org/10.1006/aama.1995.1017
  23. E.D. Rainville, Special Functions. The Macmillan Company, New York, 1960.
  24. K.A.M. Sayyed, M.S. Metwally and R.S. Batahan, On generalized Hermite matrix polynomials, Electron. J. Linear Algebra, 10 (2003), 272-279.
  25. K.A.M. Sayyed, M.S. Metwally and R.S. Batahan, Gegenbauer matrix polynomials and second order matrix differential equations, Divulgaciones Matematicas, 12 (2004), 101-115.
  26. A. Shehata, A new extension of Hermite-Hermite matrix polynomials and their properties, Thai Journal of Mathematics, 10 (2) (2012), 433-444.
  27. A. Shehata and R. Bhukya, Some properties of Hermite matrix polynomials, Journal of Inte. Math. Virtual institute, 5 (2015), 1-17.
  28. A. Shehata, On new extensions of the generalized Hermite matrix polynomials, Acta et Commentationes Universitatis Tartuensis de Mathematica, 22 (2) (2018), 203-222. https://doi.org/10.12697/ACUTM.2018.22.17
  29. A. Shehata, Certain properties of generalized Hermite-Type matrix polynomials using Weisners group theoretic techniques. Bulletin of the Brazilian Mathematical Society, New Series, 50 (2019), 419-434. https://doi.org/10.1007/s00574-018-0108-6
  30. V. Singh, M. A.Khan, A.H. Khan, On a Multivariable Extension of Laguerre Matrix Ploynomials. Electronic Journal of Mathematical Analysis and Applications, 6(1) (2018), 223-240.
  31. V. Singh, M. A.Khan and A.H. Khan, Study of Gegenbauer Matrix Polynomials via Matrix functions And Their Properties. Asian Journal of Mathematics and computer research, 16(4) (2017), 197-207.
  32. V. Singh, M. A.Khan and A.H. Khan, A note on Modified Hermite Matrix Polynomials, communicated for publication.
  33. V. Singh, M. A.Khan and A.H. Khan, Two variable Modified Hermite Matrix polynomials properties and its applications, Accepted for publication in American Institute of Physics (AIP) conference proceeding.
  34. V. Singh, M. A.Khan and A.H. Khan, Analytical properties of (q, k)-Mittag Leer function, Afr. Mat. (2019). https://doi.org/10.1007/s13370-019-00750-8.
  35. H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Ellis Horwood Limited, 1984.