Gauss quadrature formula: An extension via interpolating orthogonal polynomials

The n -point Gauss quadrature rule states that ∫ - 1 1 f ( x ) ω ( x ) d x = ∑ i = 1 n w i f ( z i ) + R n ( f ) , where z i and w i , i = 1 , … , n , are called, respectively, the Gaussian nodes and weights. It is known that the formula is exact of degree 2 n - 1 . We provide an extension of this rule by considering x = - 1 and 1 as the pre-assigned nodes of certain order n 1 and n 2 , respectively. For this, we construct interpolating orthogonal polynomials that make the suggested rule capable of utilizing the maximum information related to the value and derivatives of the integrand f at these points. Our proposed rule is different from Gauss–Lobatto and Gauss–Radau quadrature formulae, which also take care of these points to a certain extent. The results related to the degree of exactness and convergence are also presented. Some questions related to our proposed rule which may require further investigation are narrated as well.