New quadrature formulas based on the zeros of Jacobi polynomials

Abstract

The main object of this paper is to construct a new quadrature formula based on the zeros of the polynomial (1 − x 2 ) 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 .