Remark on the formula by Rakhmanov and Steklov's conjecture (Q259098)

The Steklov conjecture dates back to 1921 and it asks whether a sequence of orthonormal polynomials \((P_n)_n\) with respect to a weight function \(\rho\) is bounded at any point \(x\in(-1, 1)\) provided that \(\rho(x)\) is positive on \([-1, 1]\). The answer, that was negative, was given by Rahmanov in 1979 who constructed a weight function \(\rho(x)\) on the interval \([-1,1]\) such that \(\rho(x)\geq\delta > 0\), \(x\in[-1,1]\), whereas the corresponding sequence of orthonormal polynomials is unbounded at \(0\) [\textit{E. A. Rakhmanov}, Math. USSR, Sb. 36, 549--575 (1980); translation from Mat. Sb., Nov. Ser. 108(150), 581--608 (1979; Zbl 0452.33012)]. In the present paper the author gives a different construction of the polynomials found by Rakhmanov in his paper by using another approach based on the method developed in [\textit{A. Aptekarev} et al., J. Am. Math. Soc. 29, No. 4, 1117--1165 (2016; Zbl 1348.42024)].