Discrete Painlevé equations for recurrence coefficients of semiclassical Laguerre polynomials
HTML articles powered by AMS MathViewer
- by Lies Boelen and Walter Van Assche PDF
- Proc. Amer. Math. Soc. 138 (2010), 1317-1331 Request permission
Abstract:
We consider two semiclassical extensions of the Laguerre weight and their associated sets of orthogonal polynomials. These polynomials satisfy a three-term recurrence relation. We show that the coefficients appearing in this relation satisfy discrete Painlevé equations.References
- M. P. Bellon and C.-M. Viallet, Algebraic entropy, Comm. Math. Phys. 204 (1999), no. 2, 425–437. MR 1704282, DOI 10.1007/s002200050652
- Yang Chen and Mourad E. H. Ismail, Ladder operators for $q$-orthogonal polynomials, J. Math. Anal. Appl. 345 (2008), no. 1, 1–10. MR 2422628, DOI 10.1016/j.jmaa.2008.03.031
- T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
- A. S. Fokas, A. R. It⋅s, and A. V. Kitaev, Discrete Painlevé equations and their appearance in quantum gravity, Comm. Math. Phys. 142 (1991), no. 2, 313–344. MR 1137067, DOI 10.1007/BF02102066
- Géza Freud, On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Roy. Irish Acad. Sect. A 76 (1976), no. 1, 1–6. MR 419895
- A. Ramani, B. Grammaticos, and J. Hietarinta, Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67 (1991), no. 14, 1829–1832. MR 1125951, DOI 10.1103/PhysRevLett.67.1829
- A. Ramani and B. Grammaticos, Discrete Painlevé equations: coalescences, limits and degeneracies, Phys. A 228 (1996), no. 1-4, 160–171. MR 1399286, DOI 10.1016/0378-4371(95)00439-4
- B. Grammaticos and A. Ramani, Discrete Painlevé equations: a review, Discrete integrable systems, Lecture Notes in Phys., vol. 644, Springer, Berlin, 2004, pp. 245–321. MR 2087743, DOI 10.1007/978-3-540-40357-9_{7}
- B. Grammaticos, A. Ramani, and V. Papageorgiou, Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67 (1991), no. 14, 1825–1828. MR 1125950, DOI 10.1103/PhysRevLett.67.1825
- M. E. H. Ismail, Z. Mansour, $q$-Analogues of Freud weights and nonlinear difference equations, manuscript.
- Alphonse P. Magnus, Freud’s equations for orthogonal polynomials as discrete Painlevé equations, Symmetries and integrability of difference equations (Canterbury, 1996) London Math. Soc. Lecture Note Ser., vol. 255, Cambridge Univ. Press, Cambridge, 1999, pp. 228–243. MR 1705232, DOI 10.1017/CBO9780511569432.019
- F. W. Nijhoff, On a $q$-deformation of the discrete Painlevé $\textrm {I}$ equation and $q$-orthogonal polynomials, Lett. Math. Phys. 30 (1994), no. 4, 327–336. MR 1271093, DOI 10.1007/BF00751068
- A. Ramani, B. Grammaticos, and T. Tamizhmani, Quadratic relations in continuous and discrete Painlevé equations, J. Phys. A 33 (2000), no. 15, 3033–3044. MR 1766506, DOI 10.1088/0305-4470/33/15/310
- Walter Van Assche, Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials, Difference equations, special functions and orthogonal polynomials, World Sci. Publ., Hackensack, NJ, 2007, pp. 687–725. MR 2451211, DOI 10.1142/9789812770752_{0}058
Additional Information
- Lies Boelen
- Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, BE-3001 Leuven, Belgium
- Email: lies.boelen@wis.kuleuven.be
- Walter Van Assche
- Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, BE-3001 Leuven, Belgium
- MR Author ID: 176825
- ORCID: 0000-0003-3446-6936
- Email: walter@wis.kuleuven.be
- Received by editor(s): February 23, 2009
- Received by editor(s) in revised form: July 2, 2009
- Published electronically: December 8, 2009
- Additional Notes: This research was supported by K. U. Leuven Research Grant OT/08/033, FWO Research Grant G.0427.09 and the Belgian Interuniversity Attraction Poles Programme P6/02.
- Communicated by: Peter A. Clarkson
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 1317-1331
- MSC (2010): Primary 39A13, 33C45; Secondary 42C05
- DOI: https://doi.org/10.1090/S0002-9939-09-10152-1
- MathSciNet review: 2578525