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Asymptotics of Polynomials Orthogonal with respect to a Logarithmic Weight
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight $w(x)\mathrm{d}x = \log \frac{2k}{1-x}\mathrm{d}x$ on $(-1,1)$, $k > 1$, and verify a conjecture of A. Magnus for these coefficients. We use Riemann–Hilbert/steepest-descent methods, but not in the standard way as there is no known parametrix for the Riemann–Hilbert problem in a neighborhood of the logarithmic singularity at $x=1$.
Keywords: orthogonal polynomials; Riemann–Hilbert problems; recurrence coefficients; steepest descent method. 
@article{SIGMA_2018_14_a55,
     author = {Thomas Oliver Conway and Percy Deift},
     title = {Asymptotics of {Polynomials} {Orthogonal} with respect to a {Logarithmic} {Weight}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2018},
     volume = {14},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a55/}
}
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Thomas Oliver Conway; Percy Deift. Asymptotics of Polynomials Orthogonal with respect to a Logarithmic Weight. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a55/

[1] Abramowitz M., Stegun I. A. (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992 | MR

[2] Breuer J., Simon B., Zeitouni O., Large deviations and sum rules for spectral theory, arXiv: 1608.01467

[3] Calderón A.P., “Commutators of singular integral operators”, Proc. Nat. Acad. Sci. USA, 53 (1965), 1092–1099 | DOI | MR | Zbl

[4] Clancey K. F., Gohberg I., “Factorization of matrix functions and singular integral operators”, Operator Theory: Advances and Applications, 3, Birkhäuser Verlag, Basel–Boston, Mass., 1981 | DOI | MR | Zbl

[5] Coifman R. R., McIntosh A., Meyer Y., “L'intégrale de Cauchy définit un opérateur borné sur $L^{2}$ pour les courbes lipschitziennes”, Ann. of Math., 116 (1982), 361–387 | DOI | MR | Zbl

[6] David G., “Courbes corde-arc et espaces de Hardy généralisés”, Ann. Inst. Fourier (Grenoble), 32 (1982), 227–239 | DOI | MR | Zbl

[7] Deift P., Orthogonal polynomials and random matrices: a Riemann–Hilbert approach, Courant Lecture Notes in Mathematics, 3, New York University, Courant Institute of Mathematical Sciences, New York; Amer. Math. Soc., Providence, RI, 1999 | MR

[8] Deift P., Its A., Krasovsky I., “Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: some history and some recent results”, Comm. Pure Appl. Math., 66 (2013), 1360–1438, arXiv: 1207.4990 | DOI | MR | Zbl

[9] Deift P., Kriecherbauer T., McLaughlin K. T.-R., Venakides S., Zhou X., “Strong asymptotics of orthogonal polynomials with respect to exponential weights”, Comm. Pure Appl. Math., 52 (1999), 1491–1552 | 3.3.CO;2-R class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[10] Deift P., Zhou X., “Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space”, Comm. Pure Appl. Math., 56 (2003), 1029–1077, arXiv: math.AP/0206222 | DOI | MR | Zbl

[11] Fokas A. S., Its A. R., Kitaev A. V., “The isomonodromy approach to matrix models in $2$D quantum gravity”, Comm. Math. Phys., 147 (1992), 395–430 | DOI | MR | Zbl

[12] Gradshteyn I. S., Ryzhik I. M., Table of integrals, series, and products, 8th ed., Elsevier/Academic Press, Amsterdam, 2015 | MR | Zbl

[13] Kuijlaars A. B. J., McLaughlin K. T.-R., Van Assche W., Vanlessen M., “The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$”, Adv. Math., 188 (2004), 337–398, arXiv: math.CA/0111252 | DOI | MR | Zbl

[14] Magnus A., Gaussian integation formulas for logarithmic weights and appliction to 2-dimensional solid-state lattices, Version from August 20, 2016 https://perso.uclouvain.be/alphonse.magnus/grapheneR2.pdf

[15] Szegő G., “On certain Hermitian forms associated with the Fourier series of a positive function”, Comm. Sém. Math. Univ. Lund, 1952, suppl., 228–238 | MR | Zbl

[16] Van Assche W., “Multiple orthogonal polynomials, irrationality and transcendence”, Continued Fractions: from Analytic Number Theory to Constructive Approximation (Columbia, MO, 1998), Contemp. Math., 236, Amer. Math. Soc., Providence, RI, 1999, 325–342 | DOI | MR | Zbl

[17] Venakides S., “The solution of completely integrable systems in the continuum limit of the spectral data”, Oscillation Theory, Computation, and Methods of Compensated Compactness (Minneapolis, Minn., 1985), IMA Vol. Math. Appl., 2, Springer, New York, 1986, 337–355 | DOI | MR