Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 056, 66 pages      arXiv:1711.01590
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

Asymptotics of Polynomials Orthogonal with respect to a Logarithmic Weight

Thomas Oliver Conway and Percy Deift
Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Str., New York, NY 10012, USA

Received November 29, 2017, in final form May 30, 2018; Published online June 12, 2018

In this paper we compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight $w(x){\rm d}x = \log \frac{2k}{1-x}{\rm d}x$ 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 $x=1$.

Key words: orthogonal polynomials; Riemann-Hilbert problems; recurrence coefficients; steepest descent method.

pdf (755 kb)   tex (48 kb)


  1. Abramowitz M., Stegun I.A. (Editors), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992.
  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.
  4. Clancey K.F., Gohberg I., Factorization of matrix functions and singular integral operators, Operator Theory: Advances and Applications, Vol. 3, Birkhäuser Verlag, Basel - Boston, Mass., 1981.
  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.
  6. David G., Courbes corde-arc et espaces de Hardy généralisés, Ann. Inst. Fourier (Grenoble) 32 (1982), 227-239.
  7. Deift P., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, Vol. 3, New York University, Courant Institute of Mathematical Sciences, New York, Amer. Math. Soc., Providence, RI, 1999.
  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.
  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.
  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, math.AP/0206222.
  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.
  12. Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, 8th ed., Elsevier/Academic Press, Amsterdam, 2015.
  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, math.CA/0111252.
  14. Magnus A., Gaussian integation formulas for logarithmic weights and appliction to 2-dimensional solid-state lattices, Version from August 20, 2016,
  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.
  16. Van Assche W., Multiple orthogonal polynomials, irrationality and transcendence, in Continued Fractions: from Analytic Number Theory to Constructive Approximation (Columbia, MO, 1998), Contemp. Math., Vol. 236, Amer. Math. Soc., Providence, RI, 1999, 325-342.
  17. Venakides S., The solution of completely integrable systems in the continuum limit of the spectral data, in Oscillation Theory, Computation, and Methods of Compensated Compactness (Minneapolis, Minn., 1985), IMA Vol. Math. Appl., Vol. 2, Springer, New York, 1986, 337-355.

Previous article  Next article   Contents of Volume 14 (2018)