Polynomials orthogonal w.r.t. the logarithmic weight Recently, I have encountered the family of orthogonal polynomials $p_{n}(x)$ which is orthogonal w.r.t. the function $-\ln(x)$ on $(0,1)$. This means we have
$$\int_{0}^{1}p_{n}(x)p_{m}(x)\ln(1/x)dx=\delta_{mn}, \quad \forall m,n\geq0.$$
Most likely, this family of orthogonal polynomials have been studied in past since the weight function is very simple. Nevertheless, I found no really useful information about them till now. Primarily, I am interested in the coefficients from the three-term recurrence relation but any other properties related to $p_{n}(x)$ would be valuable. It seems this family does to match any of the well known and described family of orthogonal polynomials (belonging to the Askey scheme).
Any relevant information on literature is desirable. Thank you.
 A: As you perhaps know already, there are several systematic methods that in principle recover the recursion coefficients from the orthogonality measure $d\mu(x)=-\chi_{(0,1)}(x)\ln x\, dx$. However, it is usually safe to assume that this will not give a very explicit answer.
Here, you could for example compute the moments
$$
m_n = \int x^n\, d\mu(x) = \frac{1}{(n+1)^2} ,
$$
and then there are formulae that express $a_n,b_n$ in terms of determinants of Hankel type matrices built from these moments. See for example Teschl's book on Jacobi matrices; see formulae (2.109), (2.113), and (2.118). As expected, these seem to be getting out of hand quickly.
W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3  (1982), 289–317 presents some numerical work on this question for several weight functions, including the one $w(x)=-\ln x$ you are interested in.
A: 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, also mentioned in the comment by F. Petrov).

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.''
