Ak. Varma et E. Landau, NEW QUADRATURE-FORMULAS BASED ON THE ZEROS OF JACOBI-POLYNOMIALS, Computers & mathematics with applications, 30(3-6), 1995, pp. 213-220
The main object of this paper is to construct a new quadrature formula
based on the zeros of the polynomial (1 - x(2))P-n((alpha,beta))(x)P-
n((alpha,beta)') (x), where P-n((alpha,beta)) (x) is the Jacobi polyno
mial of degree n. It is interesting to mention that this quadrature fo
rmula is closely related to the well-known Gaussian Quadrature formula
, and above all the coefficients are also nonnegative. Thus, the quadr
ature formula stated in Theorem 1 converges to integral(-1)(1) f(x)(1
- x)(alpha)(1 + x)(beta) dx.