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

SIGMA 14 (2018), 088, 19 pages      arXiv:1804.02856
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI

Galina Filipuk a and Walter Van Assche b
a) Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland
b) Department of Mathematics, KU Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium

Received April 10, 2018, in final form August 20, 2018; Published online August 24, 2018

We investigate the recurrence coefficients of discrete orthogonal polynomials on the non-negative integers with hypergeometric weights and show that they satisfy a system of non-linear difference equations and a non-linear second order differential equation in one of the parameters of the weights. The non-linear difference equations form a pair of discrete Painlevé equations and the differential equation is the $\sigma$-form of the sixth Painlevé equation. We briefly investigate the asymptotic behavior of the recurrence coefficients as $n\to \infty$ using the discrete Painlevé equations.

Key words: discrete orthogonal polynomials; hypergeometric weights; discrete Painlevé equations; Painlevé VI.

pdf (419 kb)   tex (55 kb)


  1. Boelen L., Filipuk G., Smet C., Van Assche W., Zhang L., The generalized Krawtchouk polynomials and the fifth Painlevé equation, J. Difference Equ. Appl. 19 (2013), 1437-1451, arXiv:1204.5070.
  2. Boelen L., Filipuk G., Van Assche W., Recurrence coefficients of generalized Meixner polynomials and Painlevé equations, J. Phys. A: Math. Theor. 44 (2011), 035202, 19 pages.
  3. Chen Y., Zhang L., Painlevé VI and the unitary Jacobi ensembles, Stud. Appl. Math. 125 (2010), 91-112, arXiv:0911.5636.
  4. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
  5. Clarkson P.A., Painlevé equations - nonlinear special functions, in Orthogonal Polynomials and Special Functions, Lecture Notes in Math., Vol. 1883, Springer, Berlin, 2006, 331-411.
  6. Clarkson P.A., Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations, J. Phys. A: Math. Theor. 46 (2013), 185205, 18 pages, arXiv:1301.2396.
  7. Dai D., Zhang L., Painlevé VI and Hankel determinants for the generalized Jacobi weight, J. Phys. A: Math. Theor. 43 (2010), 055207, 14 pages, arXiv:0908.0558.
  8. Dominici D., Laguerre-Freud equations for generalized Hahn polynomials of type I, J. Difference Equ. Appl. 24 (2018), 916-940, arXiv:1801.02267.
  9. Dominici D., Marcellán F., Discrete semiclassical orthogonal polynomials of class one, Pacific J. Math. 268 (2014), 389-411, arXiv:1211.2005.
  10. Filipuk G., Van Assche W., Recurrence coefficients of a new generalization of the Meixner polynomials, SIGMA 7 (2011), 068, 11 pages, arXiv:1104.3773.
  11. Filipuk G., Van Assche W., Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation, Proc. Amer. Math. Soc. 141 (2013), 551-562, arXiv:1106.2959.
  12. Forrester P.J., Log-gases and random matrices, London Mathematical Society Monographs Series, Vol. 34, Princeton University Press, Princeton, NJ, 2010.
  13. Grammaticos B., Ramani A., Discrete Painlevé equations: a review, in Discrete Integrable Systems, Lecture Notes in Phys., Vol. 644, Springer, Berlin, 2004, 245-321.
  14. Hounkonnou M.N., Hounga C., Ronveaux A., Discrete semi-classical orthogonal polynomials: generalized Charlier, J. Comput. Appl. Math. 114 (2000), 361-366.
  15. 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.
  16. Ismail M.E.H., Nikolova I., Simeonov P., Difference equations and discriminants for discrete orthogonal polynomials, Ramanujan J. 8 (2004), 475-502.
  17. Kajiwara K., Noumi M., Yamada Y., Geometric aspects of Painlevé equations, J. Phys. A: Math. Theor. 50 (2017), 073001, 164 pages, arXiv:1509.08186.
  18. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  19. Lyu S., Chen Y., Exceptional solutions to the Painlevé VI equation associated with the generalized Jacobi weight, Random Matrices Theory Appl. 6 (2017), 1750003, 31 pages.
  20. Okamoto K., Polynomial Hamiltonians associated with Painlevé equations. II. Differential equations satisfied by polynomial Hamiltonians, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 367-371.
  21. Ramani A., Grammaticos B., Tamizhmani T., Tamizhmani K.M., Special function solutions of the discrete Painlevé equations, Comput. Math. Appl. 42 (2001), 603-614.
  22. Smet C., Van Assche W., Orthogonal polynomials on a bi-lattice, Constr. Approx. 36 (2012), 215-242, arXiv:1101.1817.
  23. Van Assche W., Orthogonal polynomials and Painlevé equations, Australian Mathematical Society Lecture Series, Vol. 27, Cambridge University Press, Cambridge, 2018.
  24. Van Assche W., Foupouagnigni M., Analysis of non-linear recurrence relations for the recurrence coefficients of generalized Charlier polynomials, J. Nonlinear Math. Phys. 10 (2003), suppl. 2, 231-237.

Previous article  Next article   Contents of Volume 14 (2018)