Bounds of the error of Gauss-Turán-type quadratures. II. (Q969292)

This paper is the continuation of the paper of \textit{G. V. Milovanović} and \textit{M. M. Spalević} [J. Comput. Appl. Math. 178, No. 1--2, 333--346 (2005; Zbl 1071.41028)] and is concerned with the remainder term in Gauss-Turán quadrature rule \[ \int^1_{-1} w(t) f(t)\,dt= \sum^n_{v=1} \sum^{2s}_{i=0} \lambda_{i,v}f^{(i)}(\tau_v) + R_{n,s}(f) \tag{1} \] with the weight functions \[ w_{n,\mu}(t) = (U_{n-1}(t)/n)^{2\mu+1}(1-t^2)^\mu, \] where \(\mu=l-1/2,\) \(l=1,2,\dots\) and \(U_n(\cos \Theta)=\frac{\sin(n+1)\Theta}{\sin \Theta} -\) is the Chebyshev polynomial of the second kind. An explicit representation of the remainder term in (1) on elliptic contours is obtained. Error bounds are found.