A note on the bounds of the error of Gauss–Turán-type quadratures

This note is concerned with estimates for the remainder term of the Gauss–Turán quadrature formula, R n , s ( f ) = ∫ - 1 1 w ( t ) f ( t ) d t - ∑ ν = 1 n ∑ i = 0 2 s A i , ν f ( i ) ( τ ν ) , where w ( t ) = ( U n - 1 ( t ) / n ) 2 1 - t 2 is the Gori–Michelli weight function, with U n - 1 ( t ) denoting 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 the points ± 1 and sum of semiaxes ϱ > 1 . The present paper generalizes the results in [G.V. Milovanović, M.M. Spalević, Bounds of the error of Gauss–Turán-type quadratures, J. Comput. Appl. Math. 178 (2005) 333–346], which is concerned with the same problem when s = 1 .