Abstract
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 d j-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.
Keywords
- Jacobi polynomials
- Lie algebras
- Irreducible representations
- Wigner matrices
- Operators on special functions
PACS
Mathematics Subject Classification (2000)
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
M. Berry, Why are special functions special? Phys. Today 54, 11 (2001)
G.E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge University Press, Cambridge, 1999)
G. Heckman, H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces (Academic, New York, 1994)
R. Koekoek, P.A. Lesky, R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues (Springer, Berlin, 2010) (and references therein)
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)
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. 301
L. Vinet, A. Zhedanov, A “missing” family of classical orthogonal polynomials. J. Phys. A: Math. Theor. 44, 085201 (2011)
V.X. Genest, L. Vinet, A. Zhedanov, d-Orthogonal polynomials and su(2). J. Math. Anal. Appl. 390, 472 (2012)
A. Zaghouani, Some basic d-orthogonal polynomials sets. Georgian Math. J. 12, 583 (2005)
I. Lamiri, A. Ouni, d-Orthogonality of Humbert and Jacobi type polynomials. J. Math. Anal. Appl. 341, 24 (2008)
P. Basseilhac, X. Martin, L. Vinet, A. Zhedanov, Little and big q-Jacobi polynomials and the Askey-Wilson algebra (2018). arXiv:1806.02656v2
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)
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. 46–128
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).
A.J. Durán, Constructing bispectral dual Hahn polynomials. J. Approx. Theory 189, 1–28 (2015)
E.P. Wigner, The Application of Group Theory to the Special Functions of Mathematical Physics (Princeton University Press, Princeton, 1955)
J.D. Talman, Special Functions: A Group Theoretic Approach (Benjamin, New York, 1968)
W. Miller Jr., Lie Theory and Special Functions (Academic, New York, 1968)
N.J. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, Providence, 1968)
N.J. Vilenkin, A.U. Klimyk, Representation of Lie Groups and Special Functions, vols. 1–3 (Kluwer, Dordrecht, 1991–1993) (and references therein)
N.J. Vilenkin, A.U. Klimyk, Representation of Lie Groups and Special Functions: Recent Advances (Kluwer, Dordrecht, 1995)
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)
T.H. Koornwinder, Representation of SU(2) and Jacobi polynomials (2016). arXiv:1606.08189 [math.CA]
C. Truesdell, An Essay Toward a Unified Theory of Special Functions. Annals of Mathematical Studies, vol. 18 (Princeton University Press, Princeton, 1949)
E. Celeghini, M.A. del Olmo, Coherent orthogonal polynomials. Ann. Phys. 335, 78 (2013)
E. Celeghini, M.A. del Olmo, Algebraic special functions and SO(3, 2). Ann. Phys. 333, 90 (2013)
E. Celeghini, M. Gadella, M.A. del Olmo, Spherical harmonics and rigged Hilbert spaces. J. Math. Phys. 59, 053502 (2018)
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. 133–138
E. Celeghini, M.A. del Olmo, M.A. Velasco, Lie groups, algebraic special functions and Jacobi polynomials. J. Phys.: Conf. Ser. 597, 012023 (2015)
W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, Reading, 1977)
G. Lauricella, Sulle funzioni ipergeometriche a pui variable. Rend. Circ. Mat. Palermo 7, 111 (1893)
E. Wigner, Einige Folgerungen aus der Schrödingerschen Theorie für die Termstrukturen. Z. Phys. 43, 624 (1927)
F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, NIST Handbook of Mathematical Functions (Cambridge University Press, New York, 2010)
Y.L. Luke, The Special Functions and Their Approximations, vol.1 (Academic Press, San Diego, 1969), pp. 275–276
M. Abramowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, San Diego, 1972)
L.C. Biedenharn, J.D. Louck, Angular Momentum in Quantum Mechanics (Addison-Wesley, Reading, 1981)
W.-K. Tung, Group Theory in Physics (World Scientific, Singapore, 1985)
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)
L. Infeld, T.E. Hull, The factorization method. Rev. Mod. Phys. 23, 21 (1951)
D. Fernández, J. Negro, M.A. del Olmo, Group approach to the factorization of the radial oscillator equation. Ann. Phys. 252, 386 (1996)
V. Bargmann, Irreducible unitary representations of the Lorentz group. Ann. Math. 48, 368 (1947)
Acknowledgements
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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Celeghini, E., del Olmo, M.A., Velasco, M.A. (2019). Jacobi Polynomials as su(2, 2) Unitary Irreducible Representation. In: Kuru, Ş., Negro, J., Nieto, L. (eds) Integrability, Supersymmetry and Coherent States. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-20087-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-20087-9_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-20086-2
Online ISBN: 978-3-030-20087-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)