Quadrature formulas with prescribed nodes for singular integrals (Q1874838)

The singular integral \[ S(\varphi;x)=\frac1\pi\int_{-1}^1(1-t)^\alpha(1+t)^\beta \frac{\varphi(t)}{t-x}\,dt,\quad -1<x<1,\quad \alpha,\beta>-1 \] is considered, treated in the sense the Cauchy principal value. Quadrature formulas containing the nodes \{-1,+1\} are constructed. The convergence at the endspoints is treated as well, \ even though the singular integral and the corresponding quadrature sum do not exist.