New quadrature formulas based on the zeros of Jacobi polynomials

The main object of this paper is to construct a new quadrature formula based on the zeros of the polynomial (1 − x2)P(α,β)n(x)P(α,β)′n (x), where P(α,β)n(x) is the Jacobi polynomial of degree n. It is interesting to mention that this quadrature formula is closely related to the well-known Gaussian Quadrature formula, and above all the coefficients are also nonnegative. Thus, the quadrature formula stated in Theorem 1 converges to ∫1−1 f(x)(1 − x)α(1 + x)β dx.