[1] H., Airault, Rational solutions of Painlevé equations, Stud. Appl.Math. 61 (1979), no. 1, 31–53.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[6] E., Basor, Y., Chen, T., Ehrhardt, Painlevé V and time dependent Jacobi polynomials, J. Phys. A: Math. Theor. 43 (2010), no. 1, 015204.Google Scholar

[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.Google Scholar

[9] P.M., Bleher, A., Dea˜no, Topological expansion in the cubic random matrix model, Int. Math. Res. Not. (2013), no. 12, 2699–2755.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.).Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[17] S., Bonan, P., Nevai, Orthogonal polynomials and their derivatives, I, J. Approx. Theory, 40 (1984), no. 2, 134–147.Google Scholar

[18] N., Bonneux, A.B.J., Kuijlaars, Exceptional Laguerre polynomials, arXiv:1708.03106 [math.CA] (August 2017).Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[31] T., Claeys, I., Krasovsky, Toeplitz determinants with merging singularities, Duke Math. J. 164 (2015), no. 15, 2897–2987.Google Scholar

[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.Google Scholar

[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.Google Scholar

[36] P.A., Clarkson, The third Painlevé equation and associated special polynomials, J. Phys. A: Math. Gen. 36 (2003), no. 36, 9507–9532.Google Scholar

[37] P.A., Clarkson, The fourth Painlevé equation and associated special polynomials, J. Math. Phys. 44 (2003), no. 11, 5350–5374.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.).Google Scholar

[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.Google Scholar

[43] P.A., Clarkson, K., Jordaan, A., Kelil, A generalized Freud weight, Stud. Appl. Math. 136 (2016), no. 3, 288–320.Google Scholar

[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.Google Scholar

[45] P.A., Clarkson, E.L., Mansfield, The second Painlevé equation, its hierarchy and associated special polynomials, Nonlinearity 16 (2003), no. 3, R1–R26.Google Scholar

[46] R., Conte, M., Musette, The Painlevé handbook, Springer, Dordrecht, 2008.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[55] P.A., Deift, X., Zhou, Asymptotics for the Painlevé II equation, Comm. Pure Appl. Math. 48 (1995), no. 3, 277–337.Google Scholar

[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.Google Scholar

[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.Google Scholar

[58] A.J., Dur´an, Exceptional Charlier and Hermite orthogonal polynomials, J. Approx. Theory 182 (2014), 29–58.Google Scholar

[59] A.J., Dur´an, Exceptional Meixner and Laguerre orthogonal polynomials, J. Alpprox. Theory 184 (2014), 176–208.Google Scholar

[60] A.J., Dur´an, Exceptional Hahn and Jacobi polynomials, J. Approx. Theory 214 (2017), 9–48.Google Scholar

[61] A.J., Dur´an,M. Pérez, Admissibility condition for exceptional Laguerre polynomials, J. Math. Anal. Appl. 424 (2015), no. 2, 1042–1053.Google Scholar

[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.Google Scholar

[63] G., Felder, A.D., Hemery, A.P., Veselov, Zeros ofWronskians of Hermite polynomials and Young diagrams, Physica D 241 (2012), 2131–2137.Google Scholar

[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.Google Scholar

[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.Google Scholar

[66] G., Filipuk, W., VanAssche, L., Zhang, Multiple orthogonal polynomials associated with an exponential cubic weight, J. Approx. Theory 190 (2015), 1–25.Google Scholar

[67] M.E., Fisher, R.E., Hartwig, Toeplitz determinants: some applications, theorems, and conjectures,Adv. Chem. Phys. 15 (1968), 333–353.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[83] V.I., Gromak, N.A., Lukashevich, Special classes of solutions of Painlevé's equations, Diff. Eq. 18 (1982), 317–326.Google Scholar

[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.Google Scholar

[85] J., Hietarinta, N., Joshi, F.W., Nijhoff, Discrete systems and integrability, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2016.Google Scholar

[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.Google Scholar

[88] E.L., Ince, Ordinary differential equations, Longmans, Green and Co., London, 1927; Dover Publications, New York, 1956.Google Scholar

[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.Google Scholar

[91] M.E.H., Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications 98, Cambridge University Press, 2005.Google Scholar

[92] M.E.H., Ismail, I., Nikolova, P., Simeonov, Difference equations and discriminants for discrete orthogonal polynomials, Ramanujan J. 8 (2004), 475–502.Google Scholar

[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.).Google Scholar

[97] K., Kajiwara, Y., Ohta, Determinant structure of the rational solutions for the Painlevé II equation, J. Math. Phys. 37 (1996), 4393–4704.Google Scholar

[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.Google Scholar

[99] A.A., Kapaev, Quasi-linear Stokes phenomenon for the Painlevé first equation, J. Phys. A: Math. Gen. 37 (2004), 11149–11167.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[120] M., Mazzocco, Rational solutions of the Painlevé VI equation, J. Phys. A:Math. Gen. 34 (2001), no. 11, 2281–2294.Google Scholar

[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.Google Scholar

[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.Google Scholar

[125] M., Noumi, Painlevé equations through symmetry, Translations of Mathematical Monographs, vol. 2233, Amer. Math. Soc., 2004.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[133] K., Okamoto, Studies on the Painlevé equations I. Sixth Painlevé equation PVI, Ann. Mat. Pura Appl. 146 (1987), 337–381.Google Scholar

[134] K., Okamoto, Studies on the Painlevé equations II. Fifth Painlevé equation PV, Japan J. Math. 13 (1987), 47–76.Google Scholar

[135] K., Okamoto, Studies on the Painlevé equations III. Second and fourth Painlevé equations, PII and PIV, Math. Ann. 275 (1986), 221–255.Google Scholar

[136] K., Okamoto, Studies on the Painlevé equations IV. Third Painlevé equation PIII, Funkcial. Ekvac. 30 (1987), 305–332.Google Scholar

[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].Google Scholar

[138] V., Periwal, D., Shevitz, Unitary-matrix models as exactly solvable string theories, Phys. Rev. Letters 64 (1990), 1326–1329.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar

[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.Google Scholar