Gauss-type quadrature formulas are extremely useful, efficient, and accurate methods for the numerical computation of many types of weighted integrals. However, the computation of such formulas (namely, of their weights and nodes) in closed form is generally impossible (except for a few special cases). Thus, there is a great interest in practical, usable methods for their numerical computation.

In this paper, this problem is addressed assuming that the weight function of the integral under consideration has weak algebraic singularities at both end points, and a strong algebraic singularity at some point inside the interval of integration. The required regularization of the integral is performed by a procedure similar to Hadamards finite part concept. This destroys the positivity property, and thus it is not evident that Gaussian quadrature formulas exist in this case. However, this potential problem is not discussed. Rather, formal computer algebraic methods are used to set up certain recurrence relations, which are then used to compute the required quantities.

Numerical results indicate that the method works at least in some special cases. The question of whether it always works, and why it works, remains open. No reference is made to the large number of papers addressing related problems for other weight functions that have appeared in the last 15 years, like the algorithm of Ehrich [1], and the seminal papers of Gautschi [2,3].