\(( \mathcal {R}, p,q)\)-Rogers–Szegö and Hermite Polynomials, and Induced Deformed Quantum Algebras

  • Mahouton Norbert HounkonnouEmail author
Conference paper
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


Deformed quantum algebras, namely the q-deformed algebras and their extensions to (p, q)-deformed algebras, continue to attract much attention. One of the main reasons is that these topics represent a meeting point of nowadays fast developing areas in mathematics and physics like the theory of quantum orthogonal polynomials and special functions, quantum groups, integrable systems, quantum and conformal field theories and statistics.

This contribution paper aims at characterizing the \(({\mathcal R},p,q)\)-Rogers–Szegö polynomials, and the \(({\mathcal R},p,q)\)-deformed difference equation giving rise to raising and lowering operators. These polynomials induce some realizations of generalized deformed quantum algebras, (called \(({\mathcal R},p,q)\)-deformed quantum algebras), which are here explicitly constructed. The study of continuous \(({\mathcal R},p,q)\)-Hermite polynomials is also performed. Known particular cases are recovered.


Quantum algebras \(({\mathcal R}{,\,}p{,\,}q)\)-deformed quantum algebras Orthogonal polynomials Rogers–Szegö polynomials Hermite polynomials 

Mathematics Subject Classification (2000)

Primary 33C45; Secondary 20G42 



This work is supported by TWAS Research Grant RGA No. 17-542 RG/MATHS/AF/AC_G-FR3240300147. The ICMPA-UNESCO Chair is in partnership with Daniel Iagolnitzer Foundation (DIF), France, and the Association pour la Promotion Scientifique de l’Afrique (APSA), supporting the development of mathematical physics in Africa. I am grateful to my students, Fridolin Melong and Cyrille Essossolim Haliya, who devoted their Christmas day to carefully read this manuscript and check the details.


  1. 1.
    W.A. Al-Salam, L. Carlitz, A q-analog of a formula of Toscano. Boll. Unione Matem. Ital. 12, 414–417 (1957)MathSciNetzbMATHGoogle Scholar
  2. 2.
    M. Arik, D. Coon, Hilbert spaces of analytic functions and generated coherent states. J. Math. Phys. 17, 424–427 (1976)CrossRefGoogle Scholar
  3. 3.
    N.M. Atakishiyev, S.K. Suslov, Difference analogs of the harmonic oscillator. Theor. Math. Phys. 85, 1055–1062 (1990)CrossRefGoogle Scholar
  4. 4.
    J.D. Bukweli Kyemba, M.N. Hounkonnou, Characterization of \(({\mathcal {R}},p,q)\)-deformed Rogers-Szegö polynomials: associated quantum algebras, deformed Hermite polynomials and relevant properties. J. Phys. A Math. Theor. 45, 225204 (2012)Google Scholar
  5. 5.
    I.M. Burban, A.U. Klimyk, P, Q-differentiation, P, Q-integration, and P, Q-hypergeometric functions related to quantum groups. Integr. Transform. Spec. Funct. 2, 15 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    R. Chakrabarti, R. Jagannathan, A (p, q)-oscillator realization of two-parameter quantum algebras. J. Phys. A Math. Gen. 24, L711 (1991)MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. Chari, A. Pressley, A Guide to Quantum Groups (Cambridge University Press, Cambridge, 1994)zbMATHGoogle Scholar
  8. 8.
    P. Feinsilver, Lie algebras and recursion relations III: q-analogs and quantized algebras. Acta Appl. Math. 19, 207–251 (1990)MathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Floreanini, L. Lapointe, L. Vinet, A note on (p, q)-oscillators and bibasic hypergeometric functions. J. Phys. A Math. Gen. 26, L611–L614 (1993)MathSciNetCrossRefGoogle Scholar
  10. 10.
    D. Galetti, A realization of the q-deformed harmonic oscillator: Rogers–Szegö and Stieltjes–Wigert polynomials. Braz. J. Phys. 33(1), 148–157 (2003)CrossRefGoogle Scholar
  11. 11.
    G. Gasper, M. Rahman, Basic Hypergeometric Series (Cambridge University Press, Cambridge, 1990)zbMATHGoogle Scholar
  12. 12.
    I.M. Gelfand, M.I. Graev, L. Vinet, (r, s)-hypergeometric functions of one variable. Russ. Acad. Sci. Dokl. Math. 48, 591 (1994)Google Scholar
  13. 13.
    M.N. Hounkonnou, E.B. Ngompe Nkouankam, On (p, q, μ, ν, ϕ 1, ϕ 2) generalized oscillator algebra and related bibasic hypergeometric functions. J. Phys. A Math. Theor. 40, 883543 (2007)MathSciNetGoogle Scholar
  14. 14.
    M.N. Hounkonnou, J.D. Bukweli Kyemba, Generalized \(( {\mathcal {R}},p,q)\)-deformed Heisenberg algebras: coherent states and special functions. J. Math. Phys. 51, 063518 (2010)Google Scholar
  15. 15.
    M.N. Hounkonnou, E.B. Ngompe Nkouankam, New (p, q, μ, ν, f)-deformed states. J. Phys. A Math. Theor. 40, 12113 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    E.H. Ismail Mourad, Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, vol. 98 (Cambridge University Press, Cambridge, 2005)Google Scholar
  17. 17.
    R. Jagannathan, R. Sridhar, (p, q)-Rogers–Szegö polynomials and the (p, q)-oscillator, in The Legacy of Alladi Ramakrishnan in the Mathematical Sciences (Springer, New York, 2010), pp. 491–501zbMATHGoogle Scholar
  18. 18.
    R. Jagannathan, K. Srinivasa Rao, Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series (2006). arXiv:math/0602613Google Scholar
  19. 19.
    F.H. Jackson, On q-functions and a certain difference operator. Trans. R. Soc. Edin. 46, 253–281 (1908)CrossRefGoogle Scholar
  20. 20.
    F.H. Jackson, On q-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)zbMATHGoogle Scholar
  21. 21.
    M. Jimbo, A q-difference analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–69 (1985)MathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Klimyk, K. Schmudgen, Quantum Groups and their Representation (Springer, Berlin, 1997)CrossRefGoogle Scholar
  23. 23.
    R. Koekoek, R.F. Swarttouw, The Askey-scheme of orthogonal polynomials and its q-analogue. TUDelft Report No. 98-17, 1998Google Scholar
  24. 24.
    A. Odzijewicz, Quantum algebras and q-special functions related to coherent states maps of the disc. Commun. Math. Phys. 192, 183–215 (1998 )MathSciNetCrossRefGoogle Scholar
  25. 25.
    C. Quesne, K.A. Penson, V.M. Tkachuk, Maths-type q-deformed coherent states for q > 1. Phys. Lett. A 313, 29–36 (2003)MathSciNetCrossRefGoogle Scholar
  26. 26.
    M.P. Schützenberger, Une interpretation de certaines solutions de l’equation fonctionnelle: F(x + y)F(x) + F(y). C. R. Acad. Sci. Paris 236, 352–353 (1953)MathSciNetzbMATHGoogle Scholar
  27. 27.
    G. Szegö, Collected Papers, in ed. by R. Askey, vol. 1 (Birkäuser, Basel, 1982), pp. 793–805Google Scholar
  28. 28.
    G. Szegö, in Orthogonal Polynomials. Colloquium Publications, vol. 23 (American Mathematical Society, Providence, 1991)Google Scholar

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Authors and Affiliations

  1. 1.International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair)University of Abomey-CalaviCotonouRepublic of Benin

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