|
SIGMA 14 (2018), 088, 19 pages arXiv:1804.02856
https://doi.org/10.3842/SIGMA.2018.088
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
Abstract
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)
References
-
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.
-
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.
-
Chen Y., Zhang L., Painlevé VI and the unitary Jacobi ensembles, Stud. Appl. Math. 125 (2010), 91-112, arXiv:0911.5636.
-
Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
-
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.
-
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.
-
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.
-
Dominici D., Laguerre-Freud equations for generalized Hahn polynomials of type I, J. Difference Equ. Appl. 24 (2018), 916-940, arXiv:1801.02267.
-
Dominici D., Marcellán F., Discrete semiclassical orthogonal polynomials of class one, Pacific J. Math. 268 (2014), 389-411, arXiv:1211.2005.
-
Filipuk G., Van Assche W., Recurrence coefficients of a new generalization of the Meixner polynomials, SIGMA 7 (2011), 068, 11 pages, arXiv:1104.3773.
-
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.
-
Forrester P.J., Log-gases and random matrices, London Mathematical Society Monographs Series, Vol. 34, Princeton University Press, Princeton, NJ, 2010.
-
Grammaticos B., Ramani A., Discrete Painlevé equations: a review, in Discrete Integrable Systems, Lecture Notes in Phys., Vol. 644, Springer, Berlin, 2004, 245-321.
-
Hounkonnou M.N., Hounga C., Ronveaux A., Discrete semi-classical orthogonal polynomials: generalized Charlier, J. Comput. Appl. Math. 114 (2000), 361-366.
-
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.
-
Ismail M.E.H., Nikolova I., Simeonov P., Difference equations and discriminants for discrete orthogonal polynomials, Ramanujan J. 8 (2004), 475-502.
-
Kajiwara K., Noumi M., Yamada Y., Geometric aspects of Painlevé equations, J. Phys. A: Math. Theor. 50 (2017), 073001, 164 pages, arXiv:1509.08186.
-
Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
-
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.
-
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.
-
Ramani A., Grammaticos B., Tamizhmani T., Tamizhmani K.M., Special function solutions of the discrete Painlevé equations, Comput. Math. Appl. 42 (2001), 603-614.
-
Smet C., Van Assche W., Orthogonal polynomials on a bi-lattice, Constr. Approx. 36 (2012), 215-242, arXiv:1101.1817.
-
Van Assche W., Orthogonal polynomials and Painlevé equations, Australian Mathematical Society Lecture Series, Vol. 27, Cambridge University Press, Cambridge, 2018.
-
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.
|
|