There has been recent advances in the study of orthogonal polynomials with respect to logarithmic weights of the form $$w(x)=\log\frac{2k}{(1-x)}~\text{on}~(-1,1),\qquad k>1,$$ in particular the asymptotics of their recurrence coefficients, in
T.O. Conway, P. Deift, Percy, Asymptotics of polynomials orthogonal with respect to a logarithmic weight. SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), (available here).
The proof is based on Riemann-Hilbert/steepest-descent methods and one of the main ingredient is a comparison with the Legendre orthogonal polynomials. The results verify a conjecture of A. Magnus for the recurrence coefficients.
From the paper :
``The weight $-\log x$ on $[0,1]$ corresponds to the case $k=1$ for which our analysis is not yet complete. The vanishing of the weight $\log\frac{2}{1-x}$ at the point $-1$ corresponds to a Fisher-Hartwig singularity for the related problem on the unit circle. This paper is in the line of questioning concerning the effect that singularities and zeroes in the measure have on the asymptotic behavior of orthogonal polynomials. The logarithmic singularities explored in this paper are of practical interest in both physics and mathematics.''