Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERATING FUNCTIONS FOR PHASE-SPACE HERMITE POLYNOMIALS

  • SHAIKHA K.A ALSHOMELI (Department of Mathematics, The University of Bahrain) ;
  • MOHANNAD J.S. SHAHWAN (Department of Mathematics, The University of Bahrain) ;
  • MAGED G. BIN-SAAD (Department of Mathematics, College of Eduction, University of Aden)
  • Received : 2023.12.15
  • Accepted : 2024.05.12
  • Published : 2024.05.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this work, we take advantage of Weisnerś group-theoretic and operational identities technique to establish generating functions for the phase-space Hermite polynomials of two-index and two-variables.

Keywords

Acknowledgement

We are grateful to the anonymous referees for carefully reading the manuscript and for offering valuable comments and suggestions which enabled us to improve the paper.

References

  1. P. Appell and J. Kampe de Feriet, Functions hypergeomhtriques et hypersphkiques, Polynomes d'Hermite authier Villars, Paris, 1926.
  2. N. Cotfas, J.P. Gazeau and K. Gorska, Complex and real Hermite polynomials and related quantizations, J. Phys. A. 43 (2010), 305-304.
  3. G. Dattoli, C. Chiccoli, S. Lorenzuttam, G. Maino and A. Tore, Generalized Bessel functions and generalized Hermite polynomials, J. Math. Anal. Appl. 178 (1993), 509-516.
  4. G. Dattoli, S. Lorenzutta, P.E. Ricci and C. Cesarano, On a family of hybrid polynomials, Integral Transforms and Special Functions 15 (2004), 485-490.
  5. G. Dattoli, S. Khan and G. Yasmin, Generating relations of Hermite-Tricomi functions using a representation of Lie algebra T3, Georgian Math. J. 14 (2007), 99-107.
  6. G. Dattoli, Incomplete 2D Hermite polynomials: Properties and applications, J. Math. Anal. Appl. 284 (2003), 447-454.
  7. G. Dattoli, A. Torre, S. Lorenzutta and G. Maino, Generalized forms of Bessel functions and Hermite polynomials, Annals of Numerical Math. 2 (1995), 211-232.
  8. G. Dattoli, A. Torre, Two-index two-variable Hermite-Bessel functions for synchrotron radiation in two-frequency undulators, Il Nuovo Cimento B. 112 (1997), 1557-1560.
  9. G. Dattoli, S. Lorenzutta, G. Maino and A. Torre, Theory of multi-index multivariable Bessel functions and Hermite polynomials, Le Matematiche 52 (1997), 179-197.
  10. G. Dattoli, S. Lorenzutta, G. Maino and A. Torre,Phase-space dynamics and Hermite polynomials of two variables and two indices, J. Math. Phys. 35(9) (1994), 4451-4462.
  11. G. Dattoli and A. Torre, Phase-space formalism: generalized Hermite polynomials, orthogonal functions, and creation-annihilation operators, J. Math. Phys. 36 (1995), 1636-1644.
  12. Ch. Hermite, Comptes Rendus, Oeuvres II 68 (1864), 293-308.
  13. M.E.H. Ismail and J. Zeng, A combinatorial approach to the 2D-Hermite and 2D-Laguerre polynomials, Adv. Appl. Math. 64 (2015), 70-88.
  14. Y.S. Kim and M.E. Noz, Phase-space picture of quantum mechanics, World Scientific, Singapore, 1991.
  15. W.Jr. Miller, Lie theory and special functions, Academic Press, New York, and London, 1968.
  16. I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
  17. M. Shahwan, Lie theoretic generating functions of two index generalized Hermite polynomials, South East Asian J. Math. & Math. Sc. 2 (2004), 1-7.
  18. I. Shigekawa, Eigenvalue problems for the Schrodinger operators with magneticeld on a compact Riemannian manifold, J. Funct. Anal. 75 (1987), 92-127.
  19. S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Math. Notes, Vol. 42, Princeton Univ. Press, Princeton, 1993.
  20. S.J.L. Van Eijndhoven, J.L.H. Meyers, New orthogonality relations for the Hermite polynomials and related Hilbert spaces, J. Math. Anal. Appl. 146 (1990) 89-98.
  21. L. Weisner, Group-theoretic origin of certain generating functions, Pacific J. Math. 5 (1955), 1033-1039.
  22. L. Weisner, Generating functions for Hermite functions, Canad. J. Math. 11 (1959), 141-147.
  23. A. Wunsche, Laguerre 2-D functions and their application in quantum optics, J. Phys. A. 31 (1999), 8267-8287.
  24. A. Wunsche, Transformations of Laguerre 2D-polynomials with application to quasi probabilities, J. Phys. A. 32 (1999), 3179-3199.