Bounds of the error of Gauss–Turán-type quadratures, II

This paper is concerned with bounds on the remainder term of the Gauss–Turán quadrature formula, R n , s ( f ) = ∫ − 1 1 f ( t ) w ( t ) d t − ∑ ν = 1 n ∑ i = 0 2 s λ i , ν f ( i ) ( τ ν ) , where w ( t ) = w n , ℓ ( t ) = [ U n − 1 ( t ) / n ] 2 ℓ ( 1 − t 2 ) ℓ − 1 / 2 ( ℓ ∈ N ) , U n − 1 denotes the ( n − 1 ) th degree Chebyshev polynomial of the second kind and f is a function analytic in the interior of and continuous on the boundary of an ellipse with foci at ±1 and the sum of semi-axes ϱ > 1 .