[1] H., Airault, Rational solutions of Painlevé equations, Stud. Appl.Math. 61 (1979), no. 1, 31–53.
[2] S.M., Alsulami, P., Nevai, J., Szabados, W. Van, Assche, A family of nonlinear difference equations: existence, uniqueness, and asymptotic behavior of positive solutions, J. Approx. Theory
193 (2015), 39–55.
[3] R., Askey, J., Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs of the American Mathematical Society 54, no. 319 (1985), Amer. Math. Soc., Providence, RI.
[4] M.R., Atkin, T., Claeys, F., Mezzadri, Random matrix ensembles with singularities and a hierarchy of Painlevé III equations, Int. Math. Res. Not. 2016 (2016), no. 8, 2320–2375.
[5] J., Baik, P., Deift, K., Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer.Math. Soc. 12 (1999), no. 4, 1119–1178.
[6] E., Basor, Y., Chen, T., Ehrhardt, Painlevé V and time dependent Jacobi polynomials, J. Phys. A: Math. Theor. 43 (2010), no. 1, 015204.
[7] M., Bertola, T., Bothner, Zeros of large degreeVorob'ev-Yablonski polynomials via a Hankel determinant identity, Int. Math. Res. Not. 2015, no.
19, 9330–9399.
[8] Ph., Biane, Orthogonal polynomials on the unit circle, q-gamma weights, and discrete Painlevé equations, Mosc.Math. J. 14 (2014), no. 1, 1–27.
[9] P.M., Bleher, A., Dea˜no, Topological expansion in the cubic random matrix model, Int. Math. Res. Not. (2013), no. 12, 2699–2755.
[10] P., Bleher, A., Its, Semiclassical asymptotics of orthogonal polynomials, Riemann- Hilbert problems, and universality in the random matrix model, Ann. ofMath. (2) 150 (1999), no. 1, 185–266.
[11] P., Bleher, A., Its, Double scaling limit in the random matrix model: the Riemann- Hilbert approach, Commun. Pure Appl. Math. 56 (2003), no. 4, 433–516.
[12] L., Boelen, G., Filipuk, W. Van, Assche, Recurrence coefficients of generalized Meixner polynomials and Painlevé equations, J. Phys. A: Math. Theor. 44, number 3 (2011), 035202 (19 pp.).
[13] L., Boelen, C., Smet, W. Van, Assche, q-Discrete Painlevé equations for recurrence coefficients of modified q-Freud orthogonal polynomials, J. Difference Equations Appl. 16 (2010), no. 1, 37–53.
[14] L., Boelen, W. Van, Assche, Discrete Painlevé equations for recurrencecoefficients of semiclassical Laguerre polynomials, Proc. Amer.Math. Soc. 138, no.
4 (2010), 1317–1331.
[15] L., Boelen, W. Van, Assche, Variations of Stieltjes-Wigert and q-Laguerre polynomials and their recurrence coefficients, J. Approx. Theory 193 (2015), 56–73.
[16] A., Bogatskiy, T., Claeys, A., Its, Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge, Comm. Math. Phys. 347 (2016), no. 1, 127–162.
[17] S., Bonan, P., Nevai, Orthogonal polynomials and their derivatives, I, J. Approx. Theory, 40 (1984), no. 2, 134–147.
[18] N., Bonneux, A.B.J., Kuijlaars, Exceptional Laguerre polynomials, arXiv:1708.03106 [math.CA] (August 2017).
[19] T., Bothner, P.D., Miller, Y., Sheng, Large degree asymptotics of rational solutions of the Painlevé-III equation (in preparation).
[20] L., Brightmore, F., Mezzadri, M.Y., Mo, A matrix model with singular weight and Painlevé III, Comm. Math. Phys.
333 (2015), 1317–1364.
[21] R.J., Buckingham, Large-degree asymptotics of rational Painlevé-IV functions associated to generalized Hermite polynomials, arXiv:1706.09005 [math-ph] (June 2017).
[22] R.J., Buckingham, P.D., Miller, Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour, Nonlinearity 27 (2014), no. 10, 2489–2578.
[23] R.J., Buckingham, P.D., Miller, Large-degree asymptotics of rational Painlevé-II functions: critical behaviour, Nonlinearity 28 (2015), no. 6, 1539–1596.
[24] Y., Chen, D., Dai, Painlevé V and a Pollaczek-Jacobi type orthogonal polynomials, J. Approx. Theory 162 (2010), no. 12, 2149–2167.
[25] Y., Chen, M.E.H., Ismail Ladder operators and differential equations for orthogonal polynomials, J. Phys. A: Math. Gen. 30 (1997), no. 22, 7817–7829.
[26] Y., Chen, M.E.H., Ismail Ladder operators for q-orthogonal polynomials, J.Math. Anal. Appl. 345 (2008), no. 1, 1–10.
[27] Y., Chen, A., Its, Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I, J. Approx. Theory 162 (2010), no. 2, 270–297.
[28] Y., Chen, L., Zhang, Painlevé VI and the unitary Jacobi ensembles, Stud. Math. 125 (2010), no. 1, 91–112.
[29] T., Claeys, B., Fahs, Random matrices with merging singularities and the Painlevé V equation, Symmetry Integr. Geom. Meth. Appl. (SIGMA) 12 (2016), 031, 44 pp.
[30] T., Claeys, A., Its, I., Krasovsky, Emergence of a singularity for Toeplitz determinants and Painlevé V, Duke Math. J. 160 (2011), no. 2, 207–262.
[31] T., Claeys, I., Krasovsky, Toeplitz determinants with merging singularities, Duke Math. J. 164 (2015), no. 15, 2897–2987.
[32] T., Claeys, A.B.J., Kuijlaars Universality of the double scaling limit in random matrix models, Commun. Pure Appl. Math. 59 (2006), no. 11, 1573–1603.
[33] T., Claeys, A.B.J., Kuijlaars Universality in unitary random matrix ensembles when the soft end meets the hard edge, in “Integrable Systems and RandomMatrices”, Contemp.Math. 458, Amer.Math. Soc., Providence, RI, 2008, pp. 265–279.
[34] T., Claeys, A.B.J., Kuijlaars
M., Vanlessen, Multi-critical unitary random matrix ensembles and the general Painlevé II equation, Ann. of Math. (2) 168 (2008), no. 2, 601–641.
[35] T., Claeys, M., Vanlessen, Universality of a double scaling limit near singular edge points in random matrix models, Commun. Math. Phys. 273 (2007), no. 2, 499– 532.
[36] P.A., Clarkson, The third Painlevé equation and associated special polynomials, J. Phys. A: Math. Gen. 36 (2003), no. 36, 9507–9532.
[37] P.A., Clarkson, The fourth Painlevé equation and associated special polynomials, J. Math. Phys. 44 (2003), no. 11, 5350–5374.
[38] P.A., Clarkson, Special polynomials associated with rational solutions of the fifth Painlevé equation, J. Comput. Appl. Math.
178 (2005), no. 1–2. 111–129.
[39] P.A., Clarkson, Painlevé equations— nonlinear special functions, in “Orthogonal Polynomials and Special Functions” (F., Marcell´an,W. Van, Assche, eds.), Lecture Notes in Mathematics 1883, Springer, Berlin, 2006. pp. 331–411.
[40] P.A., Clarkson, Special polynomials associated with rational and algebraic solutions of the Painlevé equations, in “Théories asymptotiques et équations de Painlevé”, Sémin. Cong. 14, Soc. Math., France, Paris, 2006, pp. 21–52.
[41] P.A., Clarkson, Recurrence coefficients for discrete orthogonal polynomials and the Painlevé equations, J. Phys. A: Math. Theor. 46 (2013), no. 18, 185205 (18 pp.).
[42] P.A., Clarkson, K., Jordaan, The relationship between semiclassical Laguerre polynomials and the fourth Painlevé equation, Constr. Approx. 39 (2014), no. 1, 223– 254.
[43] P.A., Clarkson, K., Jordaan, A., Kelil, A generalized Freud weight, Stud. Appl. Math. 136 (2016), no. 3, 288–320.
[44] P.A., Clarkson, A.F., Loureiro, W. Van, Assche, Unique positive solution for an alternative discrete Painlevé I equation, J. Difference Equations Appl. 22 (2016), no. 5, 656–675.
[45] P.A., Clarkson, E.L., Mansfield, The second Painlevé equation, its hierarchy and associated special polynomials, Nonlinearity 16 (2003), no. 3, R1–R26.
[46] R., Conte, M., Musette, The Painlevé handbook, Springer, Dordrecht, 2008.
[47] D., Dai, Asymptotics of orthogonal polynomials and the Painlevé transcendents, arXiv:1608.04513 [math.CA].
[48] D., Dai, A.B.J., Kuijlaars
Painlevé IV asymptotics for orthogonal polynomials with respect to a modified Laguerre weight, Stud. Appl. Math. 122 (2009), no. 1, 29– 83.
[49] D., Dai, L., Zhang, Painlevé VI and Hankel determinants for the generalized Jacobi weight, J. Phys. A: Math. Theor. 43 (2010), 055207, 14 pp.
[50] A., Dea˜no, D., Huybrechs, A.B.J., Kuijlaars
Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature, J. Approx. Theory 162 (2010), 2202–2224.
[51] P.A., Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, vol. 3, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.
[52] P., Deift, A., Its, I., Krasovsky, Asymptotics of Toeplitz, Jankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, Ann. of Math. (2) 174 (2011), 1243–1299.
[53] P., Deift, T., Kriecherbauer, K.T.-R., McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory 95 (1998), no. 3, 388–475.
[54] P., Deift, X., Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), 295–368.
[55] P.A., Deift, X., Zhou, Asymptotics for the Painlevé II equation, Comm. Pure Appl. Math. 48 (1995), no. 3, 277–337.
[56] D.K., Dimitrov, Y.C., Lun, Monotonicity, interlacing and electrostatic interpretation of zeros of exceptional Jacobi polynomials, J. Approx. Theory 181 (2014), 18–29.
[57] M., Duits, A.B.J., Kuijlaars
Painlevé I asymptotics for orthogonal polynomials with respect to a varying quartic weight, Nonlinearity 19 (2006), no. 10, 2211– 2245.
[58] A.J., Dur´an, Exceptional Charlier and Hermite orthogonal polynomials, J. Approx. Theory 182 (2014), 29–58.
[59] A.J., Dur´an, Exceptional Meixner and Laguerre orthogonal polynomials, J. Alpprox. Theory
184 (2014), 176–208.
[60] A.J., Dur´an, Exceptional Hahn and Jacobi polynomials, J. Approx. Theory
214 (2017), 9–48.
[61] A.J., Dur´an,M. Pérez, Admissibility condition for exceptional Laguerre polynomials, J. Math. Anal. Appl. 424 (2015), no. 2, 1042–1053.
[62] T., Ehrhardt, A status report on the asymptotic behavior of Toeplitz determinants with Fisher-Hartwig singularities, in “Recent Advances in Operator Theory (Groningen, 1998)”, Oper. Theory Adv. Appl. 124, Birkh¨auser, Basel, 2001, pp. 217–241.
[63] G., Felder, A.D., Hemery, A.P., Veselov, Zeros ofWronskians of Hermite polynomials and Young diagrams, Physica D 241 (2012), 2131–2137.
[64] G., Filipuk, W. Van, Assche, Recurrence coefficients of a new generalization of the Meixner polynomials, Symmetry Integr. Geom. Meth. Appl. (SIGMA) 7 (2011), 068.
[65] G., Filipuk, W., VanAssche, L., Zhang, The recurrencecoefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation, J. Phys. A: Math. Theor. 45, number 20 (2012), 205201, 13 pp.
[66] G., Filipuk, W., VanAssche, L., Zhang, Multiple orthogonal polynomials associated with an exponential cubic weight, J. Approx. Theory
190 (2015), 1–25.
[67] M.E., Fisher, R.E., Hartwig, Toeplitz determinants: some applications, theorems, and conjectures,Adv. Chem. Phys.
15 (1968), 333–353.
[68] A.S., Fokas, A.R., Its, A.A., Kapaev, V.Yu., Novokshenov, Painlevé transcendents: the Riemann-Hilbert approach, AMS Mathematical Surveys and Monographs, vol. 128, Amer. Math. Soc., Providence, RI, 2006.
[69] A.S., Fokas, A.R., Its, A.V., Kitaev, Discrete Painlevé equations and their appearance in quantum gravity, Commun. Math. Phys. 142 (1991), no. 2, 313–344.
[70] A.S., Fokas, A.R., Its, A.V., Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Comm.Math. Phys. 147 (1992), no. 2, 395–430.
[71] P.J., Forrester, N.S., Witte, Application of the τ-function theory of Painlevé equations to random matrices: PV, PIII, the LUE, JUE, and CUE, Comm. Pure Appl. Math. 55 (2002), no. 6, 679–727.
[72] P.J., Forrester, N.S., Witte, 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. 2004 (2004), no. 4, 160–183.
[73] P.J., Forrester, N.S., Witte, Application of the τ-function theory of Painlevé equations to random matrices: PVI, the JUE, CyUE, cJUE and scaled limits, Nagoya Math. J.
174 (2004), 29–114.
[74] P.J., Forrester, N.S., Witte, Discrete Painlevé equations for a class of PVI τ- functions given as U(N) averages,Nonlinearity
18 (2005), 2061–2088.
[75] P.J., Forrester, N.S., Witte, Bi-orthogonal polynomials on the unit circle, regular semi-classical weights and integrable systems, Constr. Approx. 24 (2006), 201– 237.
[76] M., Foupouagnigni, W. Van, Assche, Analysis of non-linear recurrence relations for the recurrence coefficients of generalized Charlier polynomials, J. Nonlinear Math. Phys. 10 Supplement
2 (2003), 231–237.
[77] G., Freud, On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Roy. Irish Acad. Sect. A 76 (1976), no. 1, 1–6.
[78] D., G´omez-Ullate, Y., Grandati, R., Milson, Rational extensions of the quantum harmonic oscillator and exceptionalHermite polynomials, J. Phys.A:Math. Gen. 47 (2014), no. 1, 015203, 27 pp.
[79] D., G´omez-Ullate, F., Marcell´an, R., Milson, Asymptotic and interlacing properties of zeros of exceptional Jacobi and Laguerre polynomials, J. Math. Anal. Appl. 399 (2013), no. 2, 480–495.
[80] B., Grammaticos, A., Ramani, Discrete Painlevé equations: a review, Lecture Notes in Physics
644 (2004), 245–321.
[81] Y., Grandati, C., Quesne, Disconjugacy, regularity of multi-index rationally extended potentials, and Laguerre exceptional polynomials, J. Math. Phys. 54 (2013), no. 7, 073512, 13 pp.
[82] V.I., Gromak, I., Laine, S., Shimomura, Painlevé differential equations in the complex plane, de Gruyter Studies in Mathematics, vol. 28, Walter de Gruyter, Berlin, 2002.
[83] V.I., Gromak, N.A., Lukashevich, Special classes of solutions of Painlevé's equations, Diff. Eq.
18 (1982), 317–326.
[84] S.P., Hastings, J.B.McLeod, A boundaryvalue problem associatedwith the second Painlevé transcendent and the Korteweg-de Vries equation, Arch. Rational Mech. Anal.
73 (1980) 31–51.
[85] J., Hietarinta, N., Joshi, F.W., Nijhoff, Discrete systems and integrability, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2016.
[86] M., Hisakado, Unitary matrix models and Painlevé III, Mod. Phys. Lett. A11 (1996), 3001–3010.
[87] M.N., Hounkonnou, C., Hounga, A., Ronveaux, Discrete semi-classical orthogonal polynomials: generalized Charlier, J. Comput. Appl.Math.
114 (2000), 361–366.
[88] E.L., Ince, Ordinary differential equations, Longmans, Green and Co., London, 1927; Dover Publications, New York, 1956.
[89] M.E.H., Ismail
The Askey–Wilson operator and summation theorems, in “Mathematical Analysis, Wavelets, and Signal Processing” (Cairo, 1994), Contemp. Math. 190, Amer. Math. Soc., Providence, RI, 1995, pp. 171–178.
[90] M.E.H., Ismail, Difference equations and quantized discriminants for qorthogonal polynomials, Adv. Appl. Math.
30 (2003), 562–589.
[91] M.E.H., Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications 98, Cambridge University Press, 2005.
[92] M.E.H., Ismail, I., Nikolova, P., Simeonov, Difference equations and discriminants for discrete orthogonal polynomials, Ramanujan J.
8 (2004), 475–502.
[93] A.R., Its, A.B.J., Kuijlaars, J. ¨Ostensson, Critical edge behavior in unitary random matrix ensembles and the thirty-fourth Painlevé transcendent, Int.Math. Res.Not. (2008), no. 9, rnn017, 67 pp.
[94] N., Joshi, A.V., Kitaev, On Boutroux's tritronquée solutions of the first Painlevé equation, Studies in Applied Mathematics
107 (2001), 253–291.
[95] K., Kajiwara, T., Masuda, On the Umemura polynomials for the Painlevé III equation, Phys. Lett. A 260 (1999), no. 6, 462–467.
[96] K., Kajiwara, M., Noumi, Y., Yamada, Geometric aspects of Painlevé equations, J. Phys. A: Math. Theor. 50 (2017), no. 7, 073001 (164 pp.).
[97] K., Kajiwara, Y., Ohta, Determinant structure of the rational solutions for the Painlevé II equation, J. Math. Phys.
37 (1996), 4393–4704.
[98] K., Kajiwara, Y., Ohta, Determinant structure of the rational solutions for the Painlevé IV equation, J. Phys. A: Math. Gen. 31 (1998), no. 10, 2431–2446.
[99] A.A., Kapaev, Quasi-linear Stokes phenomenon for the Painlevé first equation, J. Phys. A: Math. Gen.
37 (2004), 11149–11167.
[100] A.V., Kitaev, C.K., Law, J.B., McLeod, Rational solutions of the fifth Painlevé equation, Diff. Integral Equations
7 (1994), 967–1000.
[101] R., Koekoek, P.A., Lesky, R.F., Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Springer- Verlag, Berlin, 2010.
[102] A.B.J., Kuijlaars, R., McLaughlin, Generic behavior of the density of states in randommatrix theory and equilibrium problems in the presence of real analytic external fields, Comm. Pure Appl. Math. 53 (2000), no. 6, 736–785.
[103] A.B.J., Kuijlaars, K.T.-R., McLaughlin, Asymptotic zero behavior of Laguerre polynomials with negative parameter, Constr. Approx.
20 (2004), 497–523.
[104] A.B.J., Kuijlaars, R., Milson, Zeros of exceptional Hermite polynomials, J. Approx. Theory
200 (2015), 28–39.
[105] A.B.J., Kuijlaars, W. Van, Assche, Extremal polynomials on discrete sets, Proc. LondonMath. Soc. (3) 79 (1999), 191–221.
[106] E., Laguerre, Sur la réduction an fractions continues d'une fraction qui satisfait `a une équation différentielle linéaire du premier ordre dont les coefficients sont rationnels, J. Math. Pures Appl. (4) 1 (1885), 135–166.
[107] J.S., Lew, D.A., Quarles, Jr., Nonnegative solutions of a nonlinear recurrence, J. Approx. Theory
38 (1983), 357–379.
[108] D.S., Lubinsky, H.N., Mhaskar, E.B., Saff, A proof of Freud's conjecture for exponential weights, Constr. Approx. 4 (1988), no. 1, 65–83.
[109] N.A., Lukashevich, On the theory of the third Painlevé equation, Diff. Uravn. 3 (1967), no. 11, 1913–1923.(in Russian); translated in Differ. Equations 3 (1967), 994–999.
[110] S., Lyu, Y., Chen, Exceptional solutions to the Painlevé VI equation associated with the generalized Jacobi weight, Random Matrices Theory Appl. 6 (2017), no. 1, 1750003, 31 pp.
[111] A.P., Magnus, A proof of Freud's conjecture about the orthogonal polynomials related to |x|ρ exp(−x2m), for integer m, in “Orthogonal Polynomials and Applications” (Bar-le-Duc, 1984), Lecture Notes in Mathematics 1171, Springer, Berlin, 1985, pp. 362–372.
[112] A.P., Magnus, Freud's equations for orthogonal polynomials as discrete Painlevé equations, in “Symmetries and Integrability of Difference Equations”, Canterbury 1996, London Math. Soc. Lecture Note Series 255,
Cambridge University Press, 1999, pp. 228–243.
[113] A.P., Magnus, Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials, J. Comput. Appl. Math.
57 (1995), 215–237.
[114] A.P., Magnus, Special nonuniform lattice (snul) orthogonal polynomials on discrete dense sets of points, J. Comput. Appl. Math.
65 (1995), no. 1–3. 253–265.
[115] A.P., Magnus, Freud equations for Legendre polynomials on a circular arc and solution of the Gr¨unbaum-Delsarte-Janssen-Vries problem, J. Approx. Theory
139 (2006), 75–90.
[116] D., Masoero, P., Roffelsen, Poles of Painlevé IV rationals and their distribution, arXiv:1707.05222 [math.CA] (July 2017).
[117] T., Masuda, Classical transcendental solutions of the Painlevé equations and their degeneration, TohokuMath. J.
56 (2004), 467–490.
[118] T., Masuda, Y., Ohta, K., Kajiwara, A determinant formula for a class of rational solutions of Painlevé V equation, NagoyaMath. J.
168 (2002), 1–25.
[119] A., M´até, P., Nevai, T., Zaslavsky, Asymptotic expansions of ratios of coefficients of orthogonal polynomials with exponentialweights, Trans.Amer.Math. Soc. 287 (1985), no. 2, 495–505.
[120] M., Mazzocco, Rational solutions of the Painlevé VI equation, J. Phys. A:Math. Gen. 34 (2001), no. 11, 2281–2294.
[121] P.D., Miller, Y., Sheng, Rational solutions of the Painlevé-II equation revisited, Symmetry Integr. Geom. Meth. Appl. (SIGMA) 13 (2017), 065, 29 pp.
[122] P., Nevai, Orthogonal polynomials associatedwith exp(−x4), in “Second Edmonton Conference on Approximation Theory”, Canadian Math. Soc. Conf. Proc.
3 (1983), pp. 263–285.
[123] NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/,Release1.0.11of2016-06-08. Online companion to [137].
[124] A.F., Nikiforov, S.K., Suslov, V.B., Uvarov, Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991.
[125] M., Noumi, Painlevé equations through symmetry, Translations of Mathematical Monographs, vol. 2233, Amer. Math. Soc., 2004.
[126] M., Noumi, S., Okada, K., Okamoto, H., Umemura, Special polynomials associated with the Painlevé equations, II in “Integrable Systems and Algebraic Geometry” (Kobe/Kyoto, 1997), World Scientific, River Edge, NJ, 1998, pp. 349–372.
[127] M., Noumi, Y., Yamada, Umemura polynomials for the Painlevé V equation, Phys. Lett.
A247 (1998), 65–69.
[128] M., Noumi, Y., Yamada, Symmetries in the fourth Painlevé equation and Okamoto polynomials, NagoyaMath. J.
153 (1999), 53–86.
[129] V.Yu, Novokshenov, A.A., Shchelkonogov, Double scaling limit in the Painlevé IV equation and asymptotics of the Okamoto polynomials, in “Spectral theory and differential equations” (V.A. Marchenko's 90th anniversary collection), Amer. Math. Soc. Transl. Ser. 2, 233, Providence, RI, 2014, pp. 199–210.
[130] V.Yu., Novokshenov, A.A., Schelkonogov, Distribution of zeros of generalized Hermite polynomials, Ufimskii Mat. Zhurnal 7 (2015), no. 3, 57–69.(in Russian); translated in Ufa Math. J. 7 (2015), no. 3, 54–66.
[131] Y., Ohyama, H., Kawamuko, H., Sakai, K., Okamoto, Studies on the Painlevé equations, V: Third Painlevé equations of special type PIII(D7) and PIII(D8), J. Math. Sci. Univ. Tokyo
13 (2006), 145–204.
[132] K., Okamoto, Sur les feuilletages associés aux équations du second ordre `a points critiques fixés de P. Painlevé, Japan J. Math. (N.S.)
5 (1979), 1–79.
[133] K., Okamoto, Studies on the Painlevé equations I. Sixth Painlevé equation PVI, Ann. Mat. Pura Appl. 146 (1987), 337–381.
[134] K., Okamoto, Studies on the Painlevé equations II. Fifth Painlevé equation PV, Japan J. Math.
13 (1987), 47–76.
[135] K., Okamoto, Studies on the Painlevé equations III. Second and fourth Painlevé equations, PII and PIV, Math. Ann.
275 (1986), 221–255.
[136] K., Okamoto, Studies on the Painlevé equations IV. Third Painlevé equation PIII, Funkcial. Ekvac.
30 (1987), 305–332.
[137] F.W.J., Olver, D.W., Lozier, R.F., Boisvert, C.W., Clark (eds.), NIST handbook of mathematical functions, Cambridge University Press, New York, NY, 2010. Print companion to [123].
[138] V., Periwal, D., Shevitz, Unitary-matrix models as exactly solvable string theories, Phys. Rev. Letters 64 (1990), 1326–1329.
[139] I.E., Pritsker, R.S., Varga, The Szegʺo curve, zero distribution and weighted approximation, Trans.Amer.Math. Soc. 349 (1997), no. 10, 4085–4105.
[140] R., Sasaki, S., Tsujimoto, A., Zhedanov, Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations, J. Phys. A: Math. Theor.
43 (2010), no. 31, 315204, 20 pp.
[141] H., Segur, M., Ablowitz, Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent, Phys. D
3 (1981), no. 1–2. 165–184.
[142] J., Shohat, A differential equation for orthogonal polynomials, Duke Math. J. 5 (1939), no. 2, 401–417.
[143] B., Simon, Orthogonal polynomials on the unit circle, Amer. Math. Soc. Colloq. Publ. 54, Part 1 and Part 2, Amer. Math. Soc., Providence, RI, 2005.
[144] C., Smet, W. Van, Assche, Orthogonal polynomials on a bi-lattice, Constr. Approx. 36 (2012), no. 2, 215–242.
[145] G. Szegʺo, ¨Uber eine Eigenschaft der Exponentialreihe, Sitzungsber. Berl. Math. Ges.
23 (1924), 50–64.
[146] G. Szegʺo, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1939 (4th edition 1975).
[147] C.A., Tracy, H., Widom, Random unitary matrices, permutations and Painlevé, Commun. Math. Phys. 207 (1999), no. 3, 665–685.
[148] H., Umemura, Special polynomials associated with the Painlevé equations, I, manuscript (presented at the workshop on the Painlevé transcendents, Montréal, 1996).
[149] W. Van, Assche, Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials, in “Difference Equations, Special Functions and Orthogonal Polynomials” (S., Elaydi, eds.),World Scientific, 2007, pp. 687–725.
[150] W. Van, Assche, S.B., Yakubovich, Multiple orthogonal polynomials associated with Macdonald functions, Integral Transform. Spec. Funct. 9 (2000), no. 3, 229– 244.
[151] A.P., Vorobiev, On rational solutions of the second Painlevé equation, Differ. Uravn.
1 (1965), 79–81.
[152] N.S., Witte: Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy, Nagoya Math. J.
219 (2015), 127–234.
[153] S.-X., Xu, D., Dai, Y.-Q., Zhao, Critical edge behavior and the Bessel to Airy transition in the singularly perturbed Laguerre unitary ensemble, Comm. Math. Phys.
332 (2014), 1257–1296.
[154] S.-X., Xu, D., Dai, Y.-Q., Zhao, Painlevé III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight, J. Approx. Theory
192 (2015), 1–18.
[155] Shuai-Xia Xu, Yu-Qiu Zhao, Painlevé XXXIV asymptotics of orthogonal polynomials for the Gaussian weight with a jump at the edge, Stud. Appl. Math. 127 (2011), no. 1, 67–105.
[156] A.I., Yablonskii, On rational solutions of the second Painlevé equation (in Russian), Vestsi Akad. Nauvuk BSSR, Ser. Fiz. Tekh. Navuk
3 (1959), 30–35.