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

SIGMA 14 (2018), 056, 66 pages      arXiv:1711.01590      https://doi.org/10.3842/SIGMA.2018.056
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

Abstract
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.

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