Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 003, 27 pages      arXiv:1507.01821      https://doi.org/10.3842/SIGMA.2016.003

Doubling (Dual) Hahn Polynomials: Classification and Applications

Roy Oste and Joris Van der Jeugt
Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium

Received July 13, 2015, in final form January 04, 2016; Published online January 07, 2016

Abstract
We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a finite oscillator model [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our classification shows there exist three dual Hahn doubles and four Hahn doubles. The same technique is then applied to Racah polynomials, yielding also four doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set of symmetric orthogonal polynomials related to the Christoffel and Geronimus transformations. For each case, we also have an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. This extends the class of Sylvester-Kac matrices by remarkable new test matrices. We examine also the algebraic relations underlying the dual Hahn doubles, and discuss their usefulness for the construction of new finite oscillator models.

Key words: Hahn polynomial; Racah polynomial; Christoffel pair; symmetric orthogonal polynomials; tridiagonal matrix; matrix eigenvalues; finite oscillator model.

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References

  1. Atakishiev N.M., Suslov S.K., Difference analogues of the harmonic oscillator, Theoret. and Math. Phys. 85 (1990), 1055-1062.
  2. Atakishiyev N.M., Pogosyan G.S., Vicent L.E., Wolf K.B., Finite two-dimensional oscillator. I. The Cartesian model, J. Phys. A: Math. Gen. 34 (2001), 9381-9398.
  3. Atakishiyev N.M., Pogosyan G.S., Wolf K.B., Finite models of the oscillator, Phys. Part. Nuclei 36 (2005), 247-265.
  4. Bailey W.N., Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, Vol. 32, Stechert-Hafner, Inc., New York, 1964.
  5. Berezans'kiǐ Yu.M., Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, Amer. Math. Soc., Providence, R.I., 1968.
  6. Bevilacqua R., Bozzo E., The Sylvester-Kac matrix space, Linear Algebra Appl. 430 (2009), 3131-3138.
  7. Boros T., Rózsa P., An explicit formula for singular values of the Sylvester-Kac matrix, Linear Algebra Appl. 421 (2007), 407-416.
  8. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
  9. Clement P.A., A class of triple-diagonal matrices for test purposes, SIAM Rev. 1 (1959), 50-52.
  10. De Bie H., Genest V.X., Tsujimoto S., Vinet L., Zhedanov A., The Bannai-Ito algebra and some applications, J. Phys. Conf. Ser. 597 (2015), 012001, 16 pages, arXiv:1411.3913.
  11. Genest V.X., Vinet L., Zhedanov A., The algebra of dual $-1$ Hahn polynomials and the Clebsch-Gordan problem of ${\mathfrak{sl}}_{-1}(2)$, J. Math. Phys. 54 (2013), 023506, 13 pages, arXiv:1207.4220.
  12. Higham N., The test matrix toolbox for Matlab, Numerical Analysis Report No. 237, University of Manchester, 1993.
  13. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
  14. Jafarov E.I., Stoilova N.I., Van der Jeugt J., Finite oscillator models: the Hahn oscillator, J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310.
  15. Jafarov E.I., Stoilova N.I., Van der Jeugt J., The $\mathfrak{su}(2)_\alpha$ Hahn oscillator and a discrete Fourier-Hahn transform, J. Phys. A: Math. Theor. 44 (2011), 355205, 18 pages, arXiv:1106.1083.
  16. Jafarov E.I., Stoilova N.I., Van der Jeugt J., On a pair of difference equations for the ${}_4F_3$ type orthogonal polynomials and related exactly-solvable quantum systems, in Proceedings of Lie Theory and its Applications in Physics (Varna, 2013), Springer Proceedings in Mathematics & Statistics, Vol. 111, Editor V. Dobrev, Springer, Japan, 2014, 291-299, arXiv:1411.6125.
  17. Kac M., Random walk and the theory of Brownian motion, Amer. Math. Monthly 54 (1947), 369-391.
  18. Klimyk A.U., The ${\mathfrak{su}}(1,1)$-models of quantum oscillator, Ukr. J. Phys. 51 (2006), 1019-1027.
  19. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  20. Koelink H.T., Spectral theory and special functions, in Laredo Lectures on Orthogonal Polynomials and Special Functions, Editors R. Álvarez-Nodarse, F. Marcellán, W. Van Assche, Advances in the Theory of Special Functions and Orthogonal Polynomials, Nova Science Publishers, Inc., Hauppauge, NY, 2004, 45-84, math.CA/0107036.
  21. Koelink H.T., Van der Jeugt J., Convolutions for orthogonal polynomials from Lie and quantum algebra representations, SIAM J. Math. Anal. 29 (1998), 794-822, q-alg/9607010.
  22. Marcellán F., Petronilho J., Eigenproblems for tridiagonal $2$-Toeplitz matrices and quadratic polynomial mappings, Linear Algebra Appl. 260 (1997), 169-208.
  23. Nikiforov A.F., Suslov S.K., Uvarov V.B., Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991.
  24. Nomura K., Terwilliger P., Krawtchouk polynomials, the Lie algebra $\mathfrak{sl}_2$, and Leonard pairs, Linear Algebra Appl. 437 (2012), 345-375.
  25. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, Cambridge University Press, Cambridge, 2010, available at http://dlmf.nist.gov/.
  26. Slater L.J., Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966.
  27. Stoilova N.I., Van der Jeugt J., An exactly solvable spin chain related to Hahn polynomials, SIGMA 7 (2011), 033, 13 pages, arXiv:1101.4469.
  28. Sylvester J.J., Théorème sur les déterminants, Nouvelles Ann. Math. 13 (1854), 305-305.
  29. Taussky O., Todd J., Another look at a matrix of Mark Kac, Linear Algebra Appl. 150 (1991), 341-360.
  30. Terwilliger P., Introduction to Leonard pairs, J. Comput. Appl. Math. 153 (2003), 463-475.
  31. Tsujimoto S., Vinet L., Zhedanov A., Dual $-1$ Hahn polynomials: ''classical'' polynomials beyond the Leonard duality, Proc. Amer. Math. Soc. 141 (2013), 959-970, arXiv:1108.0132.
  32. Vinet L., Zhedanov A., A limit $q=-1$ for the big $q$-Jacobi polynomials, Trans. Amer. Math. Soc. 364 (2012), 5491-5507, arXiv:1011.1429.

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