Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T08:31:17.723Z Has data issue: false hasContentIssue false
  • English
  • Français

Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

Part of: Nontrigonometric harmonic analysis Infinite-dimensional Hamiltonian systems Differential equations in the complex domain Difference and functional equations Difference equations

Published online by Cambridge University Press:  11 January 2016

N. S. Witte
N. S. Witte*
Affiliation:
School of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia, n.witte@ms.unimelb.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the q-Painlevé system.

MSC classification

Secondary: 39A05: General theory 42C05: Orthogonal functions and polynomials, general theory 34M55: Painlevé and other special equations; classification, hierarchies; 34M56: Isomonodromic deformations 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 37K35: Lie-Bäcklund and other transformations
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 219 , September 2015 , pp. 127 - 234
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Álvarez-Nodarse, R. and Medem, J. C., q-classical polynomials and the q-Askey and Nikiforov-Uvarov tableaus, J. Comput. Appl. Math. 135 (2001), 197–223. MR 1850541. DOI 10.1016/S0377-0427(00)00585-9.Google Scholar
[2] Arinkin, D. and Borodin, A., Moduli spaces of d-connections and difference Painlevé equations, Duke Math. J. 134 (2006), 515–556. MR 2254625. DOI 10.1215/S0012-7094-06-13433-6.CrossRefGoogle Scholar
[3] Arinkin, D. and Borodin, A., τ-function of discrete isomonodromy transformations and probability, Com pos. Math. 145 (2009), 747–772. MR 2507747. DOI 10.1112/S0010437X08003862.Google Scholar
[4] Askey, R. and Wilson, J., Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials, Mem. Amer. Math. Soc. 54, Amer. Math. Soc., Providence, 1985. MR 0783216. DOI 10.1090/memo/0319.Google Scholar
[5] Atakishiyev, N. M. and Suslov, S. K., On the moments of classical and related polynomials, Rev. Mexicana Fís. 34 (1988), 147–151. MR 1034473.Google Scholar
[6] Atakishiyev, N. M. and Suslov, S. K., “Difference hypergeometric functions” in Progress in Approximation Theory (Tampa, 1990), Springer Ser. Comput. Math. 19, Springer, New York, 1992, 1–31. MR 1240776. DOI 10.1007/978-1-4612-2966-7_1.Google Scholar
[7] Bangerezako, G., The factorization method for the Askey–Wilson polynomials, J. Comput. Appl. Math. 107 (1999), 219–232. MR 1701532. DOI 10.1016/S0377-0427(99)00090-4.Google Scholar
[8] Bangerezako, G., The fourth order difference equation for the Laguerre-Hahn polynomials orthogonal on special non-uniform lattices, Ramanujan J. 5 (2001), 167–181. MR 1857182. DOI 10.1023/A:1011487824004.Google Scholar
[9] Bangerezako, G. and Foupouagnigni, M., Laguerre-Freud equations for the recurrence coefficients of the Laguerre-Hahn orthogonal polynomials on special non-uniform lattices, preprint, http://www.ictp.trieste.it (accessed 17 August 2015).Google Scholar
[10] Bangerezako, G. and Hounkonnou, M. N., The factorization method for the general second-order q-difference equation and the Laguerre-Hahn polynomials on the general q-lattice, J. Phys. A 36 (2003), no. 3, 765–773. MR 1959430. DOI 10.1088/0305-4470/36/3/311.Google Scholar
[11] Bangerezako, G. and Magnus, A. P., “The factorization method for the semi-classical polynomials” in Self-Similar Systems (Dubna, 1998), Joint Inst. Nuclear Res., Dubna, 1999, 295–300. MR 1819443.Google Scholar
[12] Baxter, R. J., Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982. MR 0690578.Google Scholar
[13] Belmehdi, S. and Ronveaux, A., Laguerre-Freud's equations for the recurrence coefficients of semi-classical orthogonal polynomials, J. Approx. Theory 76 (1994), 351–368. MR 1272122. DOI 10.1006/jath.1994.1022.CrossRefGoogle Scholar
[14] Biane, P., Orthogonal polynomials on the unit circle, q-gamma weights, and discrete Painlevé equations, Mosc. Math. J. 14 (2014), 1–27. MR 3221944.CrossRefGoogle Scholar
[15] Boelen, L., Smet, C., and Van Assche, W., q-Discrete Painlevé equations for recurrence coefficients of modified q-Freud orthogonal polynomials, preprint, arXiv: 0808.0982v1 [math. C A].Google Scholar
[16] Borodin, A., Discrete gap probabilities and discrete Painlevé equations, Duke Math. J. 117 (2003), 489–542. MR 1979052. DOI 10.1215/S0012-7094-03-11734-2.Google Scholar
[17] Borodin, A., Isomonodromy transformations of linear systems of difference equations, Ann. of Math. (2) 160 (2004), 1141–1182. MR 2144976. DOI 10.4007/annals.2004.160.1141.CrossRefGoogle Scholar
[18] Borodin, A. and Boyarchenko, D., Distribution of the first particle in discrete orthogonal polynomial ensembles, Comm. Math. Phys. 234 (2003), 287–338. MR 1962463. DOI 10.1007/s00220-002-0767-3.Google Scholar
[19] Chen, Y. and Ismail, M. E. H., Ladder operators and differential equations for orthogonal polynomials, J. Phys. A 30 (1997), no. 22, 7817–7829. MR 1616931. DOI 10.1088/0305-4470/30/22/020.Google Scholar
[20] Chen, Y. and Ismail, M. E. H., Ladder operators for q-orthogonal polynomials, J. Math. Anal. Appl. 345 (2008), 1–10. MR 2422628. DOI 10.1016/j.jmaa.2008.03.031.CrossRefGoogle Scholar
[21] Chen, Y., Ismail, M. E. H., and Van Assche, W., Tau-function constructions of the recurrence coefficients of orthogonal polynomials, Adv. in Appl. Math. 20 (1998), 141–168. MR 1601371. DOI 10.1006/aama.1997.0574.Google Scholar
[22] Costas-Santos, R. S. and Marcellán, F., q-Classical orthogonal polynomials: A general difference calculus approach, preprint, arXiv:math/0612097v5 [math.CA].Google Scholar
[23] Fokas, A. S., Its, A. R., and Kitaev, A. V., Discrete Painlevé equations and their appearance in quantum gravity, Comm. Math. Phys. 142 (1991), 313–344. MR 1137067.Google Scholar
[24] Fokas, A. S., Its, A. R., and Kitaev, A. V., The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), 395–430. MR 1174420.Google Scholar
[25] Forrester, P. J., Log-Gases and Random Matrices, London Math. Soc. Monogr. Ser. 34, Princeton University Press, Princeton, 2010. MR 2641363. DOI 10.1515/9781400835416.Google Scholar
[26] Forrester, P. J. and Witte, N. S., Discrete Painlevé equations, orthogonal polynomials on the unit circle, and N-recurrences for averages over U(N) —PIII' and PV τ-functions, Int. Math. Res. Not. IMRN 2004, no. 4, 160–183. MR 2040326. DOI 10.1155/S1073792804131553.Google Scholar
[27] Forrester, P. J. and Witte, N. S., Bi-orthogonal polynomials on the unit circle, regular semi-classical weights and integrable systems, Constr. Approx. 24 (2006), 201–237. MR 2239121. DOI 10.1007/s00365-005-0616-7.Google Scholar
[28] Foupouagnigni, M., On difference equations for orthogonal polynomials on nonuni-form lattices, J. Difference Equ. Appl. 14 (2008), 127–174. MR 2383000. DOI 10.1080/10236190701536199.Google Scholar
[29] Foupouagnigni, M., Hounkonnou, M. N., and Ronveaux, A., “Laguerre-Freud equations for the recurrence coefficients of Dω semi-classical orthogonal polynomials of class one” in Orthogonal Polynomials and Their Applications (Seville, 1997), J. Comput. Appl. Math. 99, 1998, 143–154. MR 1662690. DOI 10.1016/S0377-0427(98)00152-6.Google Scholar
[30] Freud, G., Orthogonal Polynomials, Pergamon Press, Oxford, 1971.Google Scholar
[31] Gasper, G. and Rahman, M., Basic Hypergeometric Series, 2nd ed., Encyclopedia Math. Appl. 96, Cambridge University Press, Cambridge, 2004. MR 2128719. DOI 10.1017/CBO9780511526251.Google Scholar
[32] Ghressi, A. and Khériji, L., The symmetrical Hq-semiclassical orthogonal polynomials of class one, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 076. MR 2529171. DOI 10.3842/SIGMA.2009.076.Google Scholar
[33] Grammaticos, B. and Ramani, A., On a novel q-discrete analogue of the Painlevé VI equation, Phys. Lett. A 257 (1999), no. 5–6, 288–292. MR 1698774. DOI 10.1016/S0375-9601(99)00296-0.Google Scholar
[34] Gupta, D. P. and Masson, D. R., “Solutions to the associated q-Askey–Wilson polynomial recurrence relation” in Approximation and Computation (West Lafayette, Ind., 1993), Internat. Ser. Numer. Math. 119, Birkhäuser, Boston, 1994, 273–284. MR 1333623.Google Scholar
[35] Hahn, W., Beiträge zur Theorie der Heineschen Reihen. Die 24 Integrale der Hyper-geometrischen q-Differenzengleichung. Das q-Analogon der Laplace-Transformation, Math. Nachr. 2 (1949), 340–379. MR 0035344.Google Scholar
[36] Hahn, W., Über Orthogonalpolynome, die q-Differenzengleichungen genügen, Math. Nachr. 2 (1949), 4–34. MR 0030647.Google Scholar
[37] Hahn, W., Über die höheren Heineschen Reihen und eine einheitliche Theorie der sogenannten speziellen Funktionen, Math. Nachr. 3 (1950), 257–294. MR 0040557.Google Scholar
[38] Hahn, W., Über uneigentliche Lösungen linearer geometrischer Differenzengleichungen, Math. Ann. 125 (1952), 67–81. MR 0051426.Google Scholar
[39] Iatrou, A. and Roberts, J. A. G., Integrable mappings of the plane preserving biquadratic invariant curves, II, Nonlinearity 15 (2002), 459–489. MR 1888861. DOI 10.1088/0951-7715/15/2/313.Google Scholar
[40] Ismail, M. E. H., “The Askey–Wilson operator and summation theorems” in Mathematical Analysis, Wavelets, and Signal Processing (Cairo, 1994), Contemp. Math. 190, Amer. Math. Soc., Providence, 1995, 171–178. MR 1354852. DOI 10.1090/conm/190/02300.Google Scholar
[41] Ismail, M. E. H., “Functional equations and electrostatic models for orthogonal polynomials” in Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publ. 40, Cambridge University Press, Cambridge, 2001, 225–244. MR 1842788.Google Scholar
[42] Ismail, M. E. H., Difference equations and quantized discriminants for q-orthogonal polynomials, Adv. Appl. Math. 30 (2003), 562–589. MR 1973957. DOI 10.1016/S01968858(02)00547-X.Google Scholar
[43] Ismail, M. E. H., Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia Math. Appl. 98, Cambridge University Press, Cambridge, 2005. MR 2191786. DOI 10.1017/CBO9781107325982.Google Scholar
[44] Ismail, M. E. H., Nikolova, I., and Simeonov, P., Difference equations and discriminants for discrete orthogonal polynomials, Ramanujan J. 8 (2004), 475–502. MR 2130522. DOI 10.1007/s11139-005-0276-z.Google Scholar
[45] Ismail, M. E. H. and Rahman, M., The associated Askey–Wilson polynomials, Trans. Amer. Math. Soc. 328 (1991), no. 1, 201–237. MR 1013333. DOI 10.2307/2001881.Google Scholar
[46] Ismail, M. E. H. and Simeonov, P., q-Difference operators for orthogonal polynomials, J. Comput. Appl. Math. 233 (2009), 749–761. MR 2583013. DOI 10.1016/j.cam.2009.02.044.Google Scholar
[47] Ismail, M. E. H. and Stanton, D., q-Taylor theorems, polynomial expansions, and interpolation of entire functions, J. Approx. Theory 123 (2003), 125–146. MR 1985020. DOI 10.1016/S0021-9045(03)00076-5.Google Scholar
[48] Ismail, M. E. H. and Witte, N. S., Discriminants and functional equations for polynomials orthogonal on the unit circle, J. Approx. Theory 110 (2001), 200–228. MR 1830537. DOI 10.1006/jath.2000.3540.Google Scholar
[49] Its, A. R., Kitaev, A. V., and Fokas, A. S., Matrix models of two-dimensional quantum gravity, and isomonodromic solutions of Painlevé “discrete equations” (in Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 187 (1991), 3–30; English translation in J. Math. Sci. 73 (1995), 415-429. MR 1111901. DOI 10.1007/BF02364564.Google Scholar
[50] Jimbo, M. and Sakai, H., A q-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145–154. MR 1403067. DOI 10.1007/BF00398316.Google Scholar
[51] Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y., and Yamada, Y., 10E9 solution to the elliptic Painlevé equation, J. Phys. A 36 (2003), no. 17, 263–272. MR 1984002. DOI 10.1088/0305-4470/36/17/102.Google Scholar
[52] Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y., and Yamada, Y., Hypergeometric solutions to the q-Painlevé equations, Int. Math. Res. Not. IMRN 2004, no. 47, 2497–2521. MR 2077840. DOI 10.1155/S1073792804140919.Google Scholar
[53] Kalnins, E. G. and Miller, W. Jr., “Symmetry techniques for q-series: Askey–Wilson polynomials” in Constructive Function Theory—86 Conference (Edmonton, 1986), Rocky Mountain J. Math. 19, 1989, 223–230. MR 1016175. DOI 10.1216/RMJ-1989-19-1-223.Google Scholar
[54] Khériji, L., An introduction to the Hq-semiclassical orthogonal polynomials, Methods Appl. Anal. 10 (2003), 387–411. MR 2059943.Google Scholar
[55] Khériji, L. and Maroni, P., The Hq-classical orthogonal polynomials, Acta Appl. Math. 71 (2002), 49–115. MR 1893361. DOI 10.1023/A:1014597619994.Google Scholar
[56] Koekoek, R. and Swarttouw, R. F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, preprint, http://aw.twi.tudelft.nl/koekoek /askey (accessed 17 August 2015).Google Scholar
[57] Koelink, H. T. and Koornwinder, T. H., “q-special functions, a tutorial” in Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, Mass., 1990), Contemp. Math. 134, Amer. Math. Soc., Providence, 1992, 141–142. MR 1187285. DOI 10.1090/conm/134/1187285.Google Scholar
[58] Koornwinder, T. H., The structure relation for Askey–Wilson polynomials, J. Comput. Appl. Math. 207 (2007), 214–226. MR 2345243. DOI 10.1016/j.cam.2006.10.015.Google Scholar
[59] Lesky, P. A., Eine Charakterisierung der klassischen kontinuierlichen-diskreten- und q-Orthogonal polynome, Shaker, Aachen, 2005.Google Scholar
[60] Magnus, A. P., “Associated Askey–Wilson polynomials as Laguerre-Hahn orthogonal polynomials” in Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math. 1329, Springer, Berlin, 1988, 261–278. MR 0973434. DOI 10.1007/BFb0083366.Google Scholar
[61] Magnus, A. P., “Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials” in Orthogonal Polynomials and Their Applications (Evian-Les-Bains, 1992), J. Comput. Appl. Math. 57, 1995, 215–237. MR 1340938. DOI 10.1016/0377-0427(93)E0247-J.Google Scholar
[62] Magnus, A. P., “Special nonuniform lattice (snul) orthogonal polynomials on discrete dense sets of points” in Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), J. Comput. Appl. Math. 65, 1995, 253–265. MR 1379135. DOI 10.1016/0377-0427(95)00114-X.Google Scholar
[63] Magnus, A. P., Elliptic hypergeometric solutions to elliptic difference equations, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 038. MR 2506174. DOI 10.3842/SIGMA.2009.038.Google Scholar
[64] Magnus, A. P., Painlevé equations for semi-classical recurrence coefficients, preprint, arXiv:math/9409228v1 [math.CA].Google Scholar
[65] Marcellán, F. and Salto, L., Discrete semi-classical orthogonal polynomials, J. Difference Equ. Appl. 4 (1998), 463–496. MR 1665164. DOI 10.1080/10236199808808156.Google Scholar
[66] Maroni, P., Une caractérisation des polynômes orthogonaux semi-classiques, C. R. Math. Acad. Sci. Paris Ser. I 301 (1985), 269–272. MR 0803215.Google Scholar
[67] Maroni, P. and Mejri, M., The I(q,ω) classical orthogonal polynomials, Appl. Numer. Math. 43 (2002), 423–458. MR 1941732. DOI 10.1016/S0168-9274(01)00180-5.Google Scholar
[68] Maroni, P. and Mejri, M., The symmetric Dω -semi-classical orthogonal polynomials of class one, Numer. Algorithms 49 (2008), 251–282. MR 2457103. DOI 10.1007/s11075 -008-9170-2.CrossRefGoogle Scholar
[69] Masuda, T., Hypergeometric τ-functions of the q-Painlevé system of type , SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 035. MR 2506177. DOI 10.3842/SIGMA.2009.035.Google Scholar
[70] Mejri, M., q-extension of some symmetrical and semi-classical orthogonal polynomials of class one, Appl. Anal. Discrete Math. 3 (2009), 78–87. MR 2499310. DOI 10.2298/AADM0901078M.Google Scholar
[71] Murata, M., Sakai, H., and Yoneda, J., Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type , J. Math. Phys. 44 (2003), 1396–1414. MR 1958273. DOI 10.1063/1.1531216.CrossRefGoogle Scholar
[72] Nijhoff, F. W., On a q-deformation of the discrete Painlevé I equation and q-orthogonal polynomials, Lett. Math. Phys. 30 (1994), 327–336. MR 1271093. DOI 10.1007/BF00751068.Google Scholar
[73] Nikiforov, A. F. and Suslov, S. K., Systems of classical orthogonal polynomials of a discrete variable on nonuniform grids (in Russian), Akad. Nauk SSSR Inst. Prikl. Mat. Preprint 1985, no. 8. MR 0794000.Google Scholar
[74] Nikiforov, A. F. and Suslov, S. K., “Classical orthogonal polynomials of a discrete variable on nonuniform lattices” in Group Theoretical Methods in Physics, Vol. I (Yurmala, 1985), VNU Sci., Utrecht, 1986, 505–511. MR 0919767.Google Scholar
[75] Nikiforov, A. F. and Suslov, S. K., Classical orthogonal polynomials of a discrete variable on nonuniform lattices, Lett. Math. Phys. 11 (1986), 27–34. MR 0824673. DOI 10.1007/BF00417461.Google Scholar
[76] Nikiforov, A. F., Suslov, S. K., and Uvarov, V. B., Construction of particular solutions for a difference equation of hypergeometric type (in Russian), Akad. Nauk SSSR Inst. Prikl. Mat. Preprint 1984, no. 142. MR 0792163.Google Scholar
[77] Nikiforov, A. F., Suslov, S. K., and Uvarov, V. B., Classical orthogonal polynomials of a discrete variable on nonuniform grids (in Russian), Dokl. Akad. Nauk SSSR 291, no. 5 (1986), 1056–1059; English translation in Soviet. Math. Dokl. 34, no. 3 (1987), 576-579. MR 0872153.Google Scholar
[78] Nikiforov, A. F., Suslov, S. K., and Uvarov, V. B., Classical Orthogonal Polynomials of a Discrete Variable (in Russian), Springer Ser. Comput. Phys., Springer, Berlin, 1991. MR 1149380. DOI 10.1007/978-3-642-74748-9.Google Scholar
[79] Ormerod, C. M., Witte, N. S., and Forrester, P. J., Connection preserving deformations and q-semi-classical orthogonal polynomials, Nonlinearity 24 (2011), 2405–2434. MR 2819929. DOI 10.1088/0951-7715/24/9/002.CrossRefGoogle Scholar
[80] Pastro, P. I., Orthogonal polynomials and some q-beta integrals of Ramanujan, J. Math. Anal. Appl. 112 (1985), 517–540. MR 0813618. DOI 10.1016/0022-247X(85)90261-6.Google Scholar
[81] Rahman, M., An integral representation of a 10φ9 and continuous bi-orthogonal 10φ9 rational functions, Canad. J. Math. 38 (1986), 605–618. MR 0845667. DOI 10.4153/CJM-1986-030-6.Google Scholar
[82] Rahman, M., q-Wilson functions of the second kind, SIAM J. Math. Anal. 17 (1986), 1280–1286. MR 0853530. DOI 10.1137/0517089.Google Scholar
[83] Rahman, M. and Suslov, S. K., Barnes and Ramanujan-type integrals on the q-linear lattice, SIAM J. Math. Anal. 25 (1994), 1002–1022. MR 1271323. DOI 10.1137/S0036141092233676.Google Scholar
[84] Rahman, M. and Suslov, S. K., The Pearson equation and the beta integrals, SIAM J. Math. Anal. 25 (1994), 646–693. MR 1266583. DOI 10.1137/S003614109222874X.Google Scholar
[85] Rains, E. M., An isomonodromy interpretation of the hypergeometric solution of the elliptic Painlevé equation, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), Paper 088. MR 2861188.Google Scholar
[86] Sakai, H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165–229. MR 1882403. DOI 10.1007/s002200100446.Google Scholar
[87] Sakai, H., A q-analog of the Garnier system, Funkcial. Ekvac. 48 (2005), 273–297. MR 2177121. DOI 10.1619/fesi.48.273.Google Scholar
[88] Sakai, H., Lax form of the q-Painlevé equation associated with the surface, J. Phys. A 39 (2006), no. 39, 12203–12210. MR 2266221. DOI 10.1088/0305-4470/39/39/S13.Google Scholar
[89] Spiridonov, V. P. and Zhedanov, A. S., “Generalized eigenvalue problem and a new family of rational functions biorthogonal on elliptic grids” in Special Functions 2000: Current Perspective and Future Directions (Tempe, 2000), NATO Sci. Ser. II Math. Phys. Chem. 30, Kluwer Acad., Dordrecht, 2001, 365–388. MR 2006295. DOI 10.1007/978-94-010-0818-114.Google Scholar
[90] Spiridonov, V. P. and Zhedanov, A. S., Elliptic grids, rational functions, and the Padé interpolation, Ramanujan J. 13 (2007), 285–310. MR 2281167. DOI 10.1007/s11139-006-0253-1.Google Scholar
[91] Suslov, S. K., On the theory of difference analogues of special functions of hypergeometric type, Uspekhi Mat. Nauk 44 (1989), 185–226; English translation in Russian Math. Surveys 44 (1989), 227–278. MR 0998364. DOI 10.1070/RM1989v044n02ABEH002045.Google Scholar
[92] Szegő, G., Orthogonal Polynomials, 3rd ed., Colloq. Publ. 23, Amer. Math. Soc., Providence, 1967. MR 0310533.Google Scholar
[93] Van Assche, W., “Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials” in Difference Equations, Special Functions and Orthogonal Polynomials, World Sci. Publ., Hackensack, N.J., 2007, 687–725. MR 2451211. DOI 10.1142/97898127707520058.CrossRefGoogle Scholar
[94] Witte, N. S., Bi-orthogonal systems on the unit circle, regular semi-classical weights and integrable systems, II, J. Approx. Theory 161 (2009), 565–616. MR 2563070. DOI 10.1016/j.jat.2008.11.017.Google Scholar
[95] Witte, N. S., Biorthogonal systems on the unit circle, regular semiclassical weights, and the discrete Garnier equations, Int. Math. Res. Not. IMRN 2009, no. 6, 988–1025. MR 2487490. DOI 10.1093/imrn/rnn152.Google Scholar
[96] Witte, N. S., Deformations of the Askey–Wilson polynomials and their Backlund transformations, preprint, 2010.Google Scholar
[97] Witte, N. S., On a family of integrals that extend the Askey–Wilson integral, J. Math. Anal. App. 421 (2015), 1101–1130. MR 3258309.Google Scholar
[98] Yamada, Y., A Lax formalism for the elliptic difference Painlevé equation, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 042. MR 2506170. DOI 10.3842/SIGMA.2009.042.Google Scholar
[99] Yamada, Y., Lax formalism for q-Painlevé equations with affine Weyl group symmetry of type Int. Math. Res. Not. IMRN 2011, no. 17, 3823–3838. MR 2836394.Google Scholar