Construction of Gaussian quadrature formulas for even weight functions

Instead of a quadrature rule of Gaussian type with respect to an even weight function on (−a, a) with n nodes, we construct the corresponding Gaussian formula on (0, a) with only [(n+1)/2] nodes. Especially, such a procedure is important in the cases of nonclassical weight functions, when the elements of the corresponding three-diagonal Jacobi matrix must be constructed numerically. In this manner, the influence of numerical instabilities in the process of construction can be significantly reduced, because the dimension of the Jacobi matrix is halved. We apply this approach to Pollaczek’s type weight functions on (−1, 1), to the weight functions on R which appear in the Abel-Plana summation processes, as well as to a class of weight functions with four free parameters, which covers the generalized ultraspherical and Hermite weights. Some numerical examples are also included.