Birkhoff quadrature formulae based on the zeros of Jacobi polynomials

The main object of this paper is to construct a Birkhoff quadrature formula of the form ∫1−1f(x)dx≈∑i-1nAif′(xi)+∑j=0kCj(fj(1)+(−1)jf(j)(−1), which is exact for the polynomials of degree ⩽ 2n + 2k + 1. We construct the formula when the nodes {Xi}1n and {xi∗}1n−1 are the zeros of the ultraspherical polynomials Pn(k)(x) and Pn(k)′(x), respectively.