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Geometric aspects of Painlevé equations. (English) Zbl 1441.34095
The paper is a survey of the geometric aspects of the Painlevé equations, with emphasis on the discrete Painlevé equations. The authors remind the geometric approach to Painlevé equations of Okamoto and Sakai, the Painlevé property, the isomonodromic deformations etc. The theory of discrete Painlevé equations relies on the geometry of the initial values space which is an eight point configuration in \(\mathbb{P}^1\times \mathbb{P}^1\) classified according to the degeneration of points. The authors explain the roles played in this theory by the affine Weyl group symmetries, of hypergeometric solutions and Lax pairs, of Picard lattices and root systems, of Bäcklund transformations and \(\tau\) functions. They provide a collection of basic data: equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

MSC:
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
14H70 Relationships between algebraic curves and integrable systems
33C20 Generalized hypergeometric series, \({}_pF_q\)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
39A10 Additive difference equations
39A13 Difference equations, scaling (\(q\)-differences)
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References:
[1] Abramowitz M and Stegun I A 1972 Handbook of Mathematical Functions: with Formulas, Graphs and Mathematical Tables (New York: Dover)
[2] Adler V E 1993 Recuttings of polygons Funct. Anal. Appl.27 141-3 · Zbl 0812.58072
[3] Arinkin D and Borodin A 2006 Moduli spaces of d-connections and difference Painlevé equations Duke Math. J.134 515-56 · Zbl 1109.39019
[4] Bailey W N 1954 Contiguous hypergeometric functions of the type 3 F 2 (1) Proc. Glasgow Math. Assoc.2 62-5 · Zbl 0056.06702
[5] Bellon M P and Viallet C M 1999 Algebraic entropy Commun. Math. Phys.204 425-37 · Zbl 0987.37007
[6] Boalch P 2009 Quivers and difference Painlevé equations Groups and Symmetries: From Neolithic Scots to John McKay(CRM Proceedings and Lecture Notes vol 47) ed J P Harnad and P Winternitz (Providence, RI: American Mathematical Society) pp 25-51 · Zbl 1182.39006
[7] Brézin E and Kazakov V A 1990 Exactly solvable theories of closed strings Phys. Lett. B 236 144-50
[8] Carstea A S and Takenawa T 2013 A note on minimization of rational surfaces obtained from birational dynamical systems J. Nonlinear Math. Phys.20 17-33 · Zbl 1420.39001
[9] Conte R (ed) 1999 The Painlevé Property: One Century Later (New York: Springer) · Zbl 0989.00036
[10] Deift P 2000 Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach(Courant Lecture Notes in Mathematics vol 3) (Providence, RI: American Mathematical Society)
[11] Douglas M R and Shenker S H 1990 Strings in less than one dimension Nucl. Phys. B 335 635-54
[12] Duistermaat J J 2010 Discrete Integrable Systems—QRT Maps and Elliptic Surfaces (New York: Springer) · Zbl 1219.14001
[13] Duistermaat J J and Joshi N 2011 Okamoto’s space for the first Painlevé equation in Boutroux coordinates Arch. Ration. Mech. Anal.202 707-85 · Zbl 1278.34102
[14] Fokas A S, Its A R, Kapaev A A and Novokshenov V Y 2006 Painlevé Transcendents: The Riemann-Hilbert Approach(Mathematical Surveys and Monographs vol 128) (Providence, RI: American Mathematical Society)
[15] Fokas A S, Its A R and Kitaev A V 1991 Discrete Painlevé equations and their appearance in quantum gravity Commun. Math. Phys.142 313-44 · Zbl 0742.35047
[16] Forrester P J 2010 Log-Gases and Random Matrices(London Mathematical Society Monographs vol 34) (Princeton, NJ: Princeton University Press)
[17] Fuchs R 1905 Sur quelques équations différentielles linéaires du second ordre C. R. Acad. Sci., Paris141 555-8 · JFM 36.0397.02
[18] Fuchs R 1907 Über lineare homogene differentialgleichungen zweiter ordnung mit drei im endlichen gelegene wesentlich singuläre stellen Math. Ann.63 301-21 · JFM 38.0362.01
[19] Fuji K and Suzuki T 2008 Higher order Painlevé system of type D 2 n + 2 (1) arising from integrable hierarchy Int. Math. Res. Not.2008 21 · Zbl 1168.34055
[20] Fuji K and Suzuki T 2010 Drinfeld-Sokolov hierarchies of type A and fourth order Painlevé systems Funkcial. Ekvac.53 143-67 · Zbl 1202.34156
[21] Gambier B 1910 Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est à points critique fixés Acta Math.33 1-55 · JFM 40.0377.02
[22] Garnier R 1912 Sur des équations différentielles du troisième ordre dont l’intégrale est uniforme et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale a ses point critiques fixés Ann. Sci.29 1-26 · JFM 43.0382.01
[23] Garnier R 1917 Etudes de l’intégrale générale de l’équation VI de M Painlevé dans le voisinage de ses singularité transcendentes Ann. Sci. Ecole Norm. Sup.34 239-353 · JFM 46.0667.04
[24] Gasper G and Rahman M 2004 Basic Hypergeometric Series(Encyclopedia of Mathematics and its Applications vol 35) (Cambridge: Cambridge University Press)
[25] Grammaticos B, Papageorgiou V and Ramani A 1997 Discrete dressing transformations and Painlevé equations Phys. Lett. A 235 475-9 · Zbl 0912.34006
[26] Grammaticos B and Ramani A 2004 Discrete Painlevé Equations: A Review, Discrete Integrable Systems(Lecture Notes in Physics vol 644) ed B Grammaticos et al (Berlin: Springer) pp 245-321 · Zbl 1064.39019
[27] Grammaticos B, Ramani A and Papageorgiou V 1991 Do integrable mappings have the Painlevé property? Phys. Rev. Lett.67 1825-8 · Zbl 0990.37518
[28] Gross D and Migdal A 1990 Nonperturbative two-dimensional quantum gravity Phys. Rev. Lett.64 127 · Zbl 1050.81610
[29] Gupta D P, Ismail M E H and Masson D R 1991 Associated continuous Hahn polynomials Can. J. Math.43 1263-80 · Zbl 0752.33002
[30] Gupta D P, Ismail M E H and Masson D R 1992 Contiguous relations, basic hypergeometric functions, and orthogonal polynomials: II. Associated big q-Jacobi polynomials J. Math. Anal. Appl.171 477-97 · Zbl 0766.33016
[31] Gupta D P and Masson D R 1998 Contiguous relations, continued fractions and orthogonality Trans. Am. Math. Soc.350 769-808 · Zbl 0887.33013
[32] Hamamoto T and Kajiwara K 2007 Hypergeometric solutions to the q-Painlevé equation of type A 4 (1) J. Phys. A: Math. Theor.40 12509-24 · Zbl 1182.39010
[33] Hamamoto T, Kajiwara K and Witte N S 2006 Hypergeometric solutions to the q-Painlevé equation of type (A 1 + A 1 ′) (1) Int. Math. Res. Not.2006 1-26
[34] Ince E L 1956 Ordinary Differential Equations (London: Dover)
[35] Iorgov N, Lisovyy O, Shchechkin A and Tykhyy Y 2014 Painlevé functions and conformal blocks Constr. Approx.39 255-72 · Zbl 1316.34096
[36] Ismail M E H and Rahman M 1991 The associated Askey-Wilson polynomials Trans. Am. Math. Soc.328 201-37 · Zbl 0738.33011
[37] Jimbo M and Miwa T 1981 Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: II Physica D 2 407-48 · Zbl 1194.34166
[38] Jimbo M and Miwa T 1981 Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: III Physica D 4 26-46 · Zbl 1194.34169
[39] Jimbo M, Miwa T, Môri Y and Sato M 1980 Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent Physica D 1 80-158 · Zbl 1194.82007
[40] Jimbo M, Miwa T and Ueno K 1981 Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function Physica D 2 306-52 · Zbl 1194.34167
[41] Jimbo M and Sakai H 1996 A q-analog of the sixth Painlevé equation Lett. Math. Phys.38 145-54 · Zbl 0859.39006
[42] Joshi N 2015 Quicksilver solutions of a q-difference first Painlevé equation Stud. Appl. Math.134 233-51 · Zbl 1316.39003
[43] Kac V G 1983 Infinite Dimensional Lie Algebras (New York: Springer Science + Business Media)
[44] Kajiwara K 2003 On a q-Painlevé III equation II: rational solutions J. Nonlinear Math. Phys.10 282-303 · Zbl 1046.39015
[45] Kajiwara K 2008 Hypergeometric solutions to additive discrete Painlevé equations with affine Weyl group symmetry of type E (in Japanese) p 3 Reports of RIAM Symp. No.19ME-S1 ‘40 years of Toda Lattice: Progress and Perspective of Studies of Nonlinear Waves’ Kyushu University
[46] Kajiwara K and Kimura K 2003 On a q-Painlevé III equation I: derivations, symmetry and Riccati type solutions J. Nonlinear Math. Phys.10 86-102 · Zbl 1028.39006
[47] Kajiwara K and Masuda T 1999 On the Umemura polynomials for the Painlevé III equation Phys. Lett. A 260 462-7 · Zbl 0939.34074
[48] Kajiwara K, Masuda T, Noumi M, Ohta Y and Yamada Y 2001 Determinant formulas for the Toda and discrete Toda equations Funkcial. Ekvac.44 291-307 (www.math.sci.kobe-u.ac.jp/∼fe/xml/mr1865393.xml) · Zbl 1145.37327
[49] Kajiwara K, Masuda T, Noumi M, Ohta Y and Yamada Y 2003 10 E 9 solution to the elliptic Painlevé equation J. Phys. A: Math. Gen.36 L263-72 · Zbl 1044.39014
[50] Kajiwara K, Masuda T, Noumi M, Ohta Y and Yamada Y 2004 Hypergeometric solutions to the q-Painlevé equations Int. Math. Res. Not.2004 2497-521 · Zbl 1076.33015
[51] Kajiwara K, Masuda T, Noumi M, Ohta Y and Yamada Y 2005 Construction of hypergeometric solutions to the q-Painlevé equations Int. Math. Res. Not.2004 1439-53 · Zbl 1082.33013
[52] Kajiwara K, Masuda T, Noumi M, Ohta Y and Yamada Y 2005 Cubic pencils and Painlevé Hamiltonians Funkcial. Ekvac.48 147-60 · Zbl 1161.34064
[53] Kajiwara K, Masuda T, Noumi M, Ohta Y and Yamada Y 2006 Point configurations, Cremona transformations and the elliptic difference Painlevé equation Séminaires et Congrès14 169-98 (www.emis.de/journals/SC/2006/14/html/smf_sem-cong_14_169-198.html) · Zbl 1137.39011
[54] Kajiwara K, Mazzocco M and Ohta Y 2007 A remark on the Hankel determinant formula for solutions of the Toda equation J. Phys. A: Math. Theor.40 12661-75 · Zbl 1129.37039
[55] Kajiwara K, Nakazono N and Tsuda T 2011 Projective reduction of the discrete Painlevé system of type (A 2 + A 1) (1) Int. Math. Res. Not.2011 930-66 · Zbl 1211.39005
[56] Kajiwara K, Noumi M and Yamada Y 2001 A study on the fourth q-Painlevé equation J. Phys. A: Math. Gen.34 8563-81 · Zbl 1002.39030
[57] Kajiwara K, Noumi M and Yamada Y 2002 Discrete dynamical systems with W (A m − 1 (1) × A n − 1 (1)) symmetry Lett. Math. Phys.60 211-9 · Zbl 1077.37046
[58] Kajiwara K, Noumi M and Yamada Y 2002 q-Painlevé systems arising from q-KP hierarchy Lett. Math. Phys.62 259-68 · Zbl 1030.37045
[59] Kajiwara K and Ohta Y 1996 Determinant structure of rational solutions for the Painlevé II equation J. Math. Phys.37 4693-704 · Zbl 0865.34010
[60] Kajiwara K and Ohta Y 1998 Determinant structure of rational solutions for the Painlevé IV equation J. Phys. A: Math. Gen.31 2431-46 · Zbl 0919.34008
[61] Kajiwara K, Ohta Y and Satsuma J 1996 Casorati determinant solutions for the discrete Painlevé III equation J. Math. Phys.37 4162-74 · Zbl 0865.34010
[62] Kajiwara K, Ohta Y, Satsuma J, Grammaticos B and Ramani A 1994 Casorati determinant solutions for the discrete Painlevé-II equation J. Phys. A: Math. Gen.27 915-22 · Zbl 0811.35150
[63] Kajiwara K, Yamamoto K and Ohta Y 1997 Rational solutions for the discrete Painlevé II equation Phys. Lett. A 232 189-99 · Zbl 1053.39500
[64] Kawai T and Takei Y 2005 Algebraic Analysis of Singular Perturbation Theory(Translation of Mathematical Monographs vol 227) (Providence, RI: American Mathematical Society)
[65] Kawakami H, Nakamura A and Sakai H 2013 Toward a classification of four-dimensional Painlevé-type equations Algebraic and Geometric Aspects of Integrable Systems and Random Matrices(Contemporary Mathematics vol 593) ed A Dzhamay et al (Providence, RI: American Mathematical Society) pp 143-161 · Zbl 1287.34080
[66] Masson D R 1991 Associated Wilson polynomials Constr. Approx.7 521-34 · Zbl 0746.33003
[67] Masuda T 2003 On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade Funkcial. Ekvac.46 121-71 · Zbl 1151.34340
[68] Masuda T 2004 Classical transcendental solutions of the Painlevé equations and their degeneration Tohoku Math. J.56 467-90 · Zbl 1087.34063
[69] Masuda T 2009 Hypergeometric τ-functions of the q-Painlevé system of type E 7 (1) SIGMA5 035 (30 pages) · Zbl 1163.33321
[70] Masuda T 2011 Hypergeometric τ-functions of the q-Painlevé system of type E 8 (1) Ramanujan J.24 1-31 · Zbl 1208.33017
[71] Masuda T 2015 A q-analogue of the higher order Painlevé type equations with the affine Weyl group symmetry of type D Funkcial. Ekvac.58 405-30 · Zbl 1344.14014
[72] Masuda T, Ohta Y and Kajiwara K 2002 A determinant formula for a class of rational solutions of Painlevé V equation Nagoya Math. J.168 1-25 (https://projecteuclid.org/euclid.nmj/1114631777) · Zbl 1052.34085
[73] Matano T, Matumiya A and Takano K 1999 On some Hamiltonian structures of Painlevé systems: II J. Math. Soc. Japan51 843-66 · Zbl 0941.34076
[74] Mumford D 2006 Tata Lectures on Theta I (Boston: Birkhäuser)
[75] Murata M 2004 New expressions for discrete Painlevé equations Funkcial. Ekvac.47 291-305 · Zbl 1119.39015
[76] Murata M 2009 Lax forms of the q-Painlevé equations J. Phys. A: Math. Theor.42 115201 · Zbl 1202.33031
[77] Murata M, Sakai H and Yoneda J 2003 Riccati solutions of discrete Painlevé equations with Weyl group symmetry of type E 8 (1) J. Math. Phys.44 1396-414 · Zbl 1062.34102
[78] Nagoya H and Yamada Y 2014 Symmetries of quantum Lax equations for the Painlevé equations Ann. Henri Poincaré15 313-44 · Zbl 1288.81062
[79] Nijhoff F, Satsuma J, Kajiwara K, Grammaticos B and Ramani A 1996 A study of the alternate discrete Painlevé II equation Inverse Problems12 697-716 · Zbl 0860.35124
[80] Nishioka S 2011 Transcendence of solutions of q-Painlevé equation of type A 6 (1) Aequat. Math.81 121-34 · Zbl 1215.12011
[81] Noumi M 2004 Painleve Equations Through Symmetry(Translations in Mathematical Monographs vol 156) (Providence, RI: American Mathematical Society)
[82] Noumi M and Okamoto K 1997 Irreducibility of the second and the fourth Painlevé equations Funkcial. Ekvac.40 139-63 (www.math.sci.kobe-u.ac.jp/∼fe/xml/mr1454468.xml) · Zbl 0881.34052
[83] Noumi M, Takano K and Yamada Y 2002 Bäcklund transformations and the manifold of Painlevé systems Funkcial. Ekvac.45 237-58 (www.math.sci.kobe-u.ac.jp/∼fe/xml/mr1948601.xml) · Zbl 1141.34355
[84] Noumi M, Tsujimoto S and Yamada Y 2013 Padé interpolation problem for elliptic Painlevé equation Symmetries, Integrable Systems and Representations(Springer Proceedings in Mathematics & Statistics vol 40) ed K Iohara et al (London: Springer) pp 463-482 · Zbl 1277.33020
[85] Noumi M and Yamada Y 1998 Umemura polynomials for the Painlevé V equation Phys. Lett. A 247 65-9 · Zbl 0946.34077
[86] Noumi M and Yamada Y 1998 Affine Weyl groups, discrete dynamical systems and Painlevé equations Commun. Math. Phys.199 281-95 · Zbl 0952.37031
[87] Noumi M and Yamada Y 1998 Higher order Painlevé equations of Type A l (1) Funkcial. Ekvac.41 483-503 (www.math.sci.kobe-u.ac.jp/∼fe/xml/mr1676885.xml) · Zbl 1140.34303
[88] Noumi M and Yamada Y 1999 Symmetries in the fourth Painlevé equation and Okamoto polynomials Nagoya Math. J.153 53-86 (https://projecteuclid.org/euclid.nmj/1114630819) · Zbl 0932.34088
[89] Noumi M and Yamada Y 2001 Birational Weyl group action arising from a nilpotent Poisson algebra Physics and Combinatorics 1999 (Nagoya) ed A N Kirillov et al (River Edge: World Scientific) · Zbl 0991.37047
[90] Ohta Y 1999 Self-dual structures of the discrete Painlevé equations (in Japanese) RIMS Kokyuroku1098 130-7 · Zbl 0951.39502
[91] Ohta Y, Kajiwara K and Satsuma J 1999 Bilinear structure and exact solutions for the discrete Painlevé I equation Proc. 2nd Workshop on Symmetries and Integrability of Difference Equations (Cambridge: Cambridge University Press) pp 206-216
[92] Ohta Y, Ramani A and Grammaticos B 2001 An affine Weyl group approach to the eight parameter discrete Painlevé equation J. Phys. A: Math. Gen.34 10523 · Zbl 1002.39029
[93] Ohyama Y, Kawamuko H, Sakai H and Okamoto K 2006 Studies on the Painlevé equations: V. Third Painlevé equations of special type P III (D7) and P III (D8) J. Math. Sci.13 145-204 (www.ms.u-tokyo.ac.jp/journal/abstract/jms130204.html)
[94] Ohyama Y and Okumura S 2006 A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations J. Phys. A: Math. Gen.39 12129 · Zbl 1116.34072
[95] Okamoto K 1979 Sur les feuilletages associés aux équations du second ordre à points critiques fixés de P Painlevé Japan. J. Math. (N.S.)5 1-79 (www.jstage.jst.go.jp/article/math1924/5/1/5_1_1/_article) · Zbl 0426.58017
[96] Okamoto K 1981 On the τ-function of the Painlevé equations Physica D 2 525-35 · Zbl 1194.34171
[97] Okamoto K 1986 Studies on the Painlevé equations: III. Second and fourth Painlevé equation, P II and P IV Math. Ann.275 221-55 · Zbl 0589.58008
[98] Okamoto K 1987 Studies on the Painlevé equations: I. Sixth Painlevé equation P VI Ann. Math. Pure Appl.146 337-81 · Zbl 0637.34019
[99] Okamoto K 1987 Studies on the Painlevé equations: II. Fifth Painlevé equation P V Japan. J. Math.13 47-76 (www.jstage.jst.go.jp/article/math1924/13/1/13_1_47/_article) · Zbl 0694.34005
[100] Okamoto K 1987 Studies on the Painlevé equations: IV. Third Painlevé equation P III Funkcial. Ekvac.30 305-32 (www.math.sci.kobe-u.ac.jp/∼fe/xml/mr0927186.xml) · Zbl 0639.58013
[101] Okamoto K 1999 The Hamiltonians associated to the Painlevé equations The Painlevé Property: One Century Later ed R Conte (New York: Springer) pp 735-787 · Zbl 1017.37032
[102] Painlevé P 1900 Mémoire sur les équations différentielles dont l’intégrale générale est uniforme Bull. Soc. Math. Phys. France28 201-61 · JFM 31.0337.03
[103] Painlevé P 1902 Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme Acta Math.21 1-85 · JFM 32.0340.01
[104] Periwal V and Shevitz D 1990 Unitary-matrix models as exactly solvable string theories Phys. Rev. Lett.64 1326-9
[105] Quispel G R W, Roberts J A G and Thompson C J 1988 Integrable mappings and soliton equations Phys. Lett. A 126 419-21 · Zbl 0679.58023
[106] Quispel G R W, Roberts J A G and Thompson C J 1989 Integrable mappings and soliton equations II Physica D 34 183-92 · Zbl 0679.58024
[107] Rains E M 2011 An isomonodromy interpretation of the hypergeometric solution of the elliptic Painlevé equation (and generalizations) SIGMA7 088 · Zbl 1244.33018
[108] Ramani A and Grammaticos B 1996 Discrete Painlevé equations: coalescence, limits and degeneracies Physica A 228 160-71 · Zbl 0912.34011
[109] Ramani A and Grammaticos B 2009 The number of discrete Painlevé equations is infinite Phys. Lett. A 373 3028-31 · Zbl 1233.34038
[110] Ramani A, Grammaticos B and Hietarinta J 1991 Discrete versions of the Painlevé equations Phys. Rev. Lett.67 1829-32 · Zbl 1050.39500
[111] Ramani A, Grammaticos B, Tamizhmani T and Tamizhmani K M 2001 Special function solutions of the discrete PainlevéÌ equations Comput. Math. Appl.42 603-14 · Zbl 0994.33500
[112] Sakai H 2001 Rational surfaces associated with affine root systems and geometry of the Painlevé equations Commun. Math. Phys.220 165-229 · Zbl 1010.34083
[113] Sakai H 2006 Lax forms of the q-Painlevé equation associated with the A 2 (1) surface J. Phys. A: Math. Gen.39 12203-10 · Zbl 1140.39314
[114] Sasano Y 2008 Coupled Painlevé VI systems in dimension four with affine Weyl group symmetry of type D 6 (1) , II RIMS Kokyuroku Bessatsu B 5 137-52 (www.kurims.kyoto-u.ac.jp/∼kenkyubu/bessatsu/open/B5/B5.html) · Zbl 1148.34059
[115] Sato M, Miwa T and Jimbo M 1979 Holonomic quantum fields II Publ. RIMS15 201-27 Kyoto Univ. · Zbl 0433.35058
[116] Schlesinger L 1912 Über eine Klasse von differentialsystemen beliebliger ordnumg mit festen kritischer punkten J. Math.141 96-145 · JFM 43.0385.01
[117] Shioda T and Takano K 1997 On some Hamiltonian structures of Painlevé systems, I Funkcial. Ekvac.40 271-91 (www.math.sci.kobe-u.ac.jp/∼fe/xml/mr1480279.xml) · Zbl 0891.34003
[118] Spiridonov V P 2005 Classical elliptic hypergeometric functions and their applications Elliptic Integrable Systems(Rokko Lectures in Mathematics vol 18) ed M Noumi and K Takasaki pp 253-287
[119] Suzuki T and Fuji K 2012 Higher order Painlevé systems of type A, Drinfeld-Sokolov hierarchies and Fuchsian systems RIMS Kokyuroku Bessatsu B 30 181-208 (www.kurims.kyoto-u.ac.jp/∼kenkyubu/bessatsu/open/B30/B30.html) · Zbl 1264.34174
[120] Tamizhmani K M, Tamizhmani T, Grammaticos B and Ramani A 2004 Special solutions for discrete Painlevé equations Discrete Integrable Systems(Lecture Notes in Physics 644) ed B Grammaticos et al (Berlin: Springer) pp 323-382 · Zbl 1064.39021
[121] Takenawa T 2003 Weyl group symmetry of type D 5 (1) in the q-Painlevé V equation Funkcial. Ekvac.46 173-86 · Zbl 1151.34341
[122] Tokihiro T, Grammaticos B and Ramani A 2002 From the continuous P V to discrete Painlevé equations J. Phys. A: Math. Gen.35 5943-50 · Zbl 1046.39013
[123] Tsuda T 2004 Integrable mappings via rational elliptic surfaces J. Phys. A: Math. Gen.37 2721-30 · Zbl 1060.14051
[124] Tsuda T 2005 Universal characters, integrable chains and the Painlevé equations Adv. Math.197 587-606 · Zbl 1095.34057
[125] Tsuda T 2006 Tau functions of q-Painlevé III and IV equations Lett. Math. Phys.75 39-47 · Zbl 1119.39018
[126] Tsuda T 2014 UC hierarchy and monodromy preserving deformation J. Reine Angew. Math.690 1-34 · Zbl 1339.37060
[127] Tsuda T, Okamoto K and Sakai H 2005 Folding transformations of the Painlevé equations Math. Ann.331 713-38 · Zbl 1073.34101
[128] Umemura H 1981 On the irreducibility of the first differential equation of Painlevé Algebraic Geometry and Commutative Algebra vol 2 (Tokyo: Kinonkuniya) pp 771-289
[129] Umemura H 1989 Irreducibility of Painlevé transcendental functions Sugaku Expositions2 231-52 · Zbl 0807.12005
[130] Umemura H 1998 Painlevé equations and classical functions Sugaku Expositions11 77100
[131] Umemura H and Watanabe H 1997 Solutions of the second and fourth Painlevé equations. I Nagoya Math. J.148 151-98 (https://projecteuclid.org/euclid.nmj/1118767032) · Zbl 0934.33029
[132] Umemura H and Watanabe H 1998 Solutions of the third Painlevé equation. I Nagoya Math. J.151 1-24 (https://projecteuclid.org/euclid.nmj/1118766573) · Zbl 0917.34004
[133] Watanabe H 1995 Solutions of the fifth Painlevé equation: I Hokkaido Math. J.24 231-67 · Zbl 0833.34005
[134] Watanabe H 1998 Birational canonical transformations and classical solutions of the sixth Painlevé equation Ann. Scuola Norm. Super. Pisa Cl. Sci.27 379-425 · Zbl 0933.34095
[135] Whittaker E T and Watson G N 1996 A Course of Modern Analysis 4th edn (New York: Cambridge University Press)
[136] Wilson J A 1977 Three term contiguous relations and some new orthogonal polynomials Padé and Rational Approximation ed E B Saff and R S Varga (New York: Academic) pp 227-232
[137] Witte N S and Ormerod C M 2012 Construction of a Lax pair for E 6 (1) q-Painlevé system SIGMA8 097 · Zbl 1383.33005
[138] Wu T T, McCoy B M, Tracy C A and Barouch E 1976 Spin – spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region Phys. Rev. B 13 316-74
[139] Yamada Y 1999 Determinant formulas for the τ-functions of the Painlevé equations of type A Nagoya Math. J.156 123-34 (https://projecteuclid.org/euclid.nmj/1114631302) · Zbl 1134.33325
[140] Yamada Y 2009 A Lax formalism for the elliptic difference Painlevé equation SIGMA5 042 (15 pages) · Zbl 1165.39018
[141] Yamada Y 2009 Padé method to Painlevé equations Funkcial. Ekvac.52 83-92 · Zbl 1177.34115
[142] Yamada Y 2011 Lax formalism for q-Painlevé equations with affine Weyl group symmetry of type E n (1) Int. Math. Res. Not.2011 3823-38 · Zbl 1233.39008
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