Error bound of Gaussian quadrature rules for certain Gegenbauer functions

In this paper we present an extension of our previous research, focusing on a method to numerically evaluate the error term in the Gaussian quadrature formula with the Legendre weight function, as discussed by Jandrlic et al. (2022). For an analytic integrand, the error term in Gaussian quadrature can be expressed as a contour integral. Consequently, determining the upper bound of the error term involves identifying the maximum value of the modulus of the kernel within the subintegral expression for the error along this contour. In our previous study, we investigated the position of this maximum point on the ellipse for Legendre polynomials. In this paper, we establish sufficient conditions for the maximum of the modulus of the kernel, which we derived analytically, to occur at one of the semi-axes for Gegenbauer polynomials. This result extends to a significantly broader case. We present an effective error estimation that we compare with the actual one. Some numerical results are presented.