WEIGHTED QUADRATURE FORMULAE OF GAUSS–CHRISTOFFEL–STANCU TYPE

. In the present paper we consider weighted integrals and develop explicit quadrature formulae of Gauss–Christoﬀel–Stancu type using simple Gaussian nodes and multiple ﬁxed nodes. Given the multiple ﬁxed nodes and their multiplicities, we present some algorithms for ﬁnding the Gaussian nodes, the co-eﬃcients and the remainders of the corresponding quadrature formulae. Several illustrative examples are presented in the case of some classical weight functions. MSC


INTRODUCTION
Let w be a nonnegative weight function assumed integrable over a bounded or unbounded interval (a, b) of the real axis.We require that all the moments of this weight function x k w(x)dx, k = 0, 1, 2, . . .exist and c 0 > 0.
Integrands in which such a weight function is present, as a multiplicative factor, appear frequently in the theory of orthogonal families of functions.
Let f be a real-valued function having continuous derivatives of whatever orders will be needed.
We suppose that we want to construct quadrature formulae, for weighted integrals, of the following form ( 1) where and R(f ) = R(w; f ), the remainder or the complimentary term, is, by definition, the difference U (f ) − F (f ).We denote by u the polynomial of the distinct nodes x k and by ω the polynomial of the multiple fixed nodes a i , namely u(x) = (x − x 1 )(x − x 2 ) . . .(x − x m ), ( 3) Here r 1 , r 2 , . . ., r s are nonnegative integers and a i are preassigned nodes, such that ω(x) ≥ 0 on the interval (a, b).

USE OF THE LAGRANGE-HERMITE INTERPOLATION
We will use a method of parameters (see D. D. Stancu [14], [15]) for constructing a general Gauss-Christoffel type quadrature rule by using simple nodes x k and preassigned multiple nodes a i .
We shall start from the Lagrange-Hermite interpolation formula corresponding to the function f , to the simple nodes x k , to the multiple nodes a i and to other nondetermined simple nodes t 1 , t 2 , . . ., t m , distinct from the other nodes.This formula has the form ( 5) The interpolating polynomials the square brackets indicating the divided difference of f on the indicating nodes; the numbers beneath the nodes designate their multiplicities.
More explicitly the interpolating polynomial can be written as follows where

CONSTRUCTION OF THE QUADRATURES BY THE METHOD OF PARAMETERS
If we multiply formula (5) by w(x) and integrate we obtain a quadrature formula of the form Because the divided difference which occurs in (8) is of order 2m + p, it follows that the quadrature formula (7) has the degree of exactness N = 2m + p − 1.
Now we want to determine the nodes x k so that we have for any values of the parameters t 1 , t 2 , . . ., t m .
Since the coefficients B h are given by the formula , where {P n } is the orthogonal family of polynomials on (a, b), with respect to the weight function w.
We mention that formula (9) was given by E. B. Christoffel [1] in the case and by G. Szegö [18] in the case w(x) = 1 and arbitrary r 1 , r 2 , . . ., r s .
As a consequence, if x k are the roots of the polynomial (9) then we get the following quadrature formula (10 It should be remarked that x k (the Gaussian nodes) can be found also by determining the relative minimum of the following function of m variables
We can see from (11) that the coefficients A k are all positive, but the coefficients C i,j are not necessarily positive.
According to the interpolation formula (6) these last coefficients can be expressed by the formula If we assume that f ∈ C 2m+p (a, b), by using the mean-value theorem of divided differences we can write the following representation of the remainder ( 13) Remark.In some special cases formula ( 9) can be simplified.For instance if a = −b, w(x) and ω(x) are even in (−b, b) and the fixed nodes are ±a 1 , ±a 2 , . . ., ±a q , 2q = r, having the orders of multiplicity the even numbers r 1 , r 2 , . . ., r q , then the determinant from ( 9) reduces to a determinant having the first row formed by the following elements P m (x), P m+2 (x), . . ., P m+r (x) and the next rows are m+r (a i ), where i = 1, 2, . . ., q; j = 0, 1, . . ., r i − 1.
We can see that when r 1 = r 2 = r we have B j = C j if j is even and where B j > 0, j = 0, 1, . . ., r − 1.
Here the coefficients of f (i) (0) are zero if i is odd.When the number of Gaussian nodes m is odd: 2k +1, then one of these will coincide with zero and we get a quadrature formula similar with the preceding one with the multiplicity of the fixed node a 2 increased by two.
3) When the polynomial of the fixed nodes is ω(x) = x 2s , a = −b, m = 2n and the weight function is even, then we can obtain a quadrature formula of the following form ( 16) because among the fixed nodes occurs also the point x = 0.
The remainder has the expression The nodes x k are the roots of the orthogonal polynomial D 2n,2s (x) corresponding to the weight function w(x)x 2s and to the bounded or unbounded interval (−a, a) (see D. D. Stancu [13]); u(x) = D 2n,2s (x) is with leading coefficient 1.
For the coefficients A k one finds the following expressions Consequently all the coefficients A k of the quadrature formula ( 16) are positive.
If we take into account that D 2n,2s (x) is a symmetrical polynomial with respect to w(x) and the interval (−a, a) and we assume that we have

we can see that we have
By applying the Christoffel-Darboux formula from the theory of orthogonal polynomials, we can deduce the following explicit expressions for the coefficients A k (see D. D. Stancu [13]): , where D 2n,2s (x) = x 2n + . . .and by using the moments of w(x) we have .

ILLUSTRATIVE EXAMPLES IN THE CASE OF SOME CLASSICAL WEIGHT FUNCTIONS
A) In the case of the weight function w(x) = (1 − x) α (1 + x) β , where α, β > −1, we have the orthogonal polynomial of Jacobi . For the calculation of the integral we construct some Gauss-Christoffel quadrature formulae.
In the particular case α = β = 0, it reduces to the known Cavalieri-Simpson formula.
If α = 1 2 , β = − 1 2 , we obtain the following quadrature formula If we consider that the polynomial of the fixed nodes is ω(x) = (1 − x 2 ) 2 , then the Gaussian nodes can be found by solving the equation where we have denoted by J m (x) the Jacobi polynomial J (α,α) m (x).For m = 3 we have

720
, we find that the corresponding Gaussian nodes are In this case we obtain the following Gauss-Christoffel-Stancu quadrature formula (10) (ξ).
If α = 1 2 we get for the Chebyshev second kind weight function the quadrature formula When the polynomial of the fixed nodes is ω(x) = x 2 , m = 3, and w(x) = (1 − x 2 ) α then we find a quadrature formula of degree of exactness eleven, namely In the paper [15] D. D. Stancu has investigated an extended generalization of the P. Turan [19] quadrature formula by using multiple fixed nodes and multiple Gaussian nodes having odd orders of multiplicities.
We mention that in the paper [16] D. D. Stancu, in collaboration with A. H. Stroud, has tabulated the values of the Gaussian nodes, the coefficients and the remainders, with 20 significant digits, for several weighted quadrature formulae using different multiple fixed nodes.

5 .
SPECIAL CASES OF FORMULA (7) 1) If −a 1 = a 2 = b (p = 2), (a, b) = (−b, b) and w(x) is an even function then we obtain the quadrature formula b −b w