Asymptotics of Polynomials Orthogonal with respect to a Logarithmic Weight

Symmetry Integrability and Geometry-Methods and Applications

article

Thomas Oliver Conway¹  Percy Deift¹ 

[1] Department of Mathematics, Courant Institute of Mathematical Sciences, New York University

关键词: orthogonal polynomials;    Riemann–Hilbert problems;    recurrence coefficients;    steepest descent method;   

DOI  :  10.3842/SIGMA.2018.056

来源: National Academy of Science of Ukraine

【 摘 要 】

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 $x=1$.

【 授权许可】

Unknown