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.