Jacobi Polynomials as su(2, 2) Unitary Irreducible Representation

  • Enrico Celeghini
  • Mariano A. del OlmoEmail author
  • Miguel A. Velasco
Part of the CRM Series in Mathematical Physics book series (CRM)


An infinite-dimensional irreducible representation of su(2, 2) is explicitly constructed in terms of ladder operators for the Jacobi polynomials \(J_{n}^{({\alpha },\beta )}(x)\) and the Wigner dj-matrices where the integer and half-integer spins j := n + (α + β)∕2 are considered together. The 15 generators of this irreducible representation are realized in terms of zero or first order differential operators and the algebraic and analytical structure of operators of physical interest discussed.


Jacobi polynomials Lie algebras Irreducible representations Wigner matrices Operators on special functions 


02.20.Sv 02.30Gp 03.65Db 

Mathematics Subject Classification (2000)

17B15 17B81 33C45 



This research is supported in part by the Ministerio de Economía y Competitividad of Spain under grant MTM2014-57129-C2-1-P and the Junta de Castilla y León (Projects VA057U16, VA137G18, and BU229P18).


  1. 1.
    M. Berry, Why are special functions special? Phys. Today 54, 11 (2001)CrossRefGoogle Scholar
  2. 2.
    G.E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge University Press, Cambridge, 1999)CrossRefGoogle Scholar
  3. 3.
    G. Heckman, H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces (Academic, New York, 1994)zbMATHGoogle Scholar
  4. 4.
    R. Koekoek, P.A. Lesky, R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues (Springer, Berlin, 2010) (and references therein)Google Scholar
  5. 5.
    E.I. Jafarov, J. Van der Jeugt, A finite oscillator model related to \(\mathfrak {sl}(2|1)\). J. Phys. A: Math. Theor. 45 275301 (2012). Discrete series representations for \(\mathfrak {sl}(2|1)\), Meixner polynomials and oscillator models, J. Phys. A: Math. Theor. 45, 485201 (2012); The oscillator model for the Lie superalgebra \(\mathfrak {sh}(2|2)\) and Charlier polynomials, J. Math. Phys. 54, 103506 (2013)Google Scholar
  6. 6.
    J. Van der Jeugt, Finite oscillator models described by the Lie superalgebra sl(2|1), in Symmetries and Groups in Contemporary Physics, ed. by C. Bai, J.-P. Gazeau, M.-L. Ge. Nankai Series in Pure, Applied Mathematics and Theoretical Physics, vol. 11 (World Scientific, Singapore, 2013), p. 301Google Scholar
  7. 7.
    L. Vinet, A. Zhedanov, A “missing” family of classical orthogonal polynomials. J. Phys. A: Math. Theor. 44, 085201 (2011)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    V.X. Genest, L. Vinet, A. Zhedanov, d-Orthogonal polynomials and su(2). J. Math. Anal. Appl. 390, 472 (2012)Google Scholar
  9. 9.
    A. Zaghouani, Some basic d-orthogonal polynomials sets. Georgian Math. J. 12, 583 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    I. Lamiri, A. Ouni, d-Orthogonality of Humbert and Jacobi type polynomials. J. Math. Anal. Appl. 341, 24 (2008)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    P. Basseilhac, X. Martin, L. Vinet, A. Zhedanov, Little and big q-Jacobi polynomials and the Askey-Wilson algebra (2018). arXiv:1806.02656v2Google Scholar
  12. 12.
    R. Floreanini, L. Vinet, Quantum algebras and q-special functions. Ann. Phys. 221, 53 (1993); On the quantum group and quantum algebra approach to q-special functions, Lett. Math. Phys. 27, 179 (1993)Google Scholar
  13. 13.
    T.H. Koornwinder, q-special functions, a tutorial, in Representations of Lie Groups and Quantum Groups, ed. by V. Baldoni, M.A. Picardello (Longman Scientific and Technical, New York, 1994), pp. 46–128zbMATHGoogle Scholar
  14. 14.
    D. Gómez-Ullate, N. Kamran, R. Milson, An extended class of orthogonal polynomials defined by a Sturm-Liouville problem. J. Math. Anal. Appl. 359, 352 (2009).MathSciNetCrossRefGoogle Scholar
  15. 15.
    A.J. Durán, Constructing bispectral dual Hahn polynomials. J. Approx. Theory 189, 1–28 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    E.P. Wigner, The Application of Group Theory to the Special Functions of Mathematical Physics (Princeton University Press, Princeton, 1955)Google Scholar
  17. 17.
    J.D. Talman, Special Functions: A Group Theoretic Approach (Benjamin, New York, 1968)zbMATHGoogle Scholar
  18. 18.
    W. Miller Jr., Lie Theory and Special Functions (Academic, New York, 1968)zbMATHGoogle Scholar
  19. 19.
    N.J. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, 1968)CrossRefGoogle Scholar
  20. 20.
    N.J. Vilenkin, A.U. Klimyk, Representation of Lie Groups and Special Functions, vols. 1–3 (Kluwer, Dordrecht, 1991–1993) (and references therein)Google Scholar
  21. 21.
    N.J. Vilenkin, A.U. Klimyk, Representation of Lie Groups and Special Functions: Recent Advances (Kluwer, Dordrecht, 1995)CrossRefGoogle Scholar
  22. 22.
    E. Celeghini, M. Gadella, M.A. del Olmo, SU(2), associated Laguerre polynomials and rigged Hilbert spaces, in Quantum Theory and Symmetries with Lie Theory and Its Applications in Physics, vol. 2, ed. by V. Dobrev, pp. 373–383 in Springer Proceedings in Mathematics & Statistics, vol. 255 (Springer, Singapore, 2018)Google Scholar
  23. 23.
    T.H. Koornwinder, Representation of SU(2) and Jacobi polynomials (2016). arXiv:1606.08189 [math.CA]Google Scholar
  24. 24.
    C. Truesdell, An Essay Toward a Unified Theory of Special Functions. Annals of Mathematical Studies, vol. 18 (Princeton University Press, Princeton, 1949)Google Scholar
  25. 25.
    E. Celeghini, M.A. del Olmo, Coherent orthogonal polynomials. Ann. Phys. 335, 78 (2013)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    E. Celeghini, M.A. del Olmo, Algebraic special functions and SO(3,  2). Ann. Phys. 333, 90 (2013)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    E. Celeghini, M. Gadella, M.A. del Olmo, Spherical harmonics and rigged Hilbert spaces. J. Math. Phys. 59, 053502 (2018)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    E. Celeghini, M.A. del Olmo, Group theoretical aspects of \(L^2(\mathbb {R}^+)\), \(L^2(\mathbb {R}^2)\) and associated Laguerre polynomials in Physical and Mathematical Aspects of Symmetries, ed. by S. Duarte, J.P. Gazeau, et al., (Springer, New York, 2018), pp. 133–138Google Scholar
  29. 29.
    E. Celeghini, M.A. del Olmo, M.A. Velasco, Lie groups, algebraic special functions and Jacobi polynomials. J. Phys.: Conf. Ser. 597, 012023 (2015)Google Scholar
  30. 30.
    W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, Reading, 1977)Google Scholar
  31. 31.
    G. Lauricella, Sulle funzioni ipergeometriche a pui variable. Rend. Circ. Mat. Palermo 7, 111 (1893)CrossRefGoogle Scholar
  32. 32.
    E. Wigner, Einige Folgerungen aus der Schrödingerschen Theorie für die Termstrukturen. Z. Phys. 43, 624 (1927)ADSCrossRefGoogle Scholar
  33. 33.
    F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, New York, 2010)zbMATHGoogle Scholar
  34. 34.
    Y.L. Luke, The Special Functions and Their Approximations, vol.1 (Academic Press, San Diego, 1969), pp. 275–276Google Scholar
  35. 35.
    M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, San Diego, 1972)zbMATHGoogle Scholar
  36. 36.
    L.C. Biedenharn, J.D. Louck, Angular Momentum in Quantum Mechanics (Addison-Wesley, Reading, 1981)zbMATHGoogle Scholar
  37. 37.
    W.-K. Tung, Group Theory in Physics (World Scientific, Singapore, 1985)CrossRefGoogle Scholar
  38. 38.
    E. Schrödinger, Further studies on solving eigenvalue problems by factorization. Proc. Roy. Irish Acad. A46, 183 (1940/1941); The Factorization of the Hypergeometric Equation, Proc. Roy. Irish Acad. A47, 53 (1941)Google Scholar
  39. 39.
    L. Infeld, T.E. Hull, The factorization method. Rev. Mod. Phys. 23, 21 (1951)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    D. Fernández, J. Negro, M.A. del Olmo, Group approach to the factorization of the radial oscillator equation. Ann. Phys. 252, 386 (1996)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    V. Bargmann, Irreducible unitary representations of the Lorentz group. Ann. Math. 48, 368 (1947)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Enrico Celeghini
    • 1
    • 2
  • Mariano A. del Olmo
    • 2
    Email author
  • Miguel A. Velasco
    • 3
  1. 1.Dpto di FisicaUniversità di Firenze and INFN–Sezione di FirenzeFirenzeItaly
  2. 2.Dpto de Física Teórica and IMUVAUniv. de ValladolidValladolidSpain
  3. 3.Departamento de Física Teórica, Atómica y ÓpticaUniversidad de ValladolidValladolidSpain

Personalised recommendations