The sequence of orthogonal polynomials satisfies a three term recurrence whose coefficients are the weighted centroids and the ratios of the weighted norms of the successive polynomials. From another perspective, it is also a multiple scales problem. The scales are the inherent scale of the weight and the polynomials and the interval length. In practice, the calculations involve integrating a 2nth degree polynomial for each nth-order coefficient and marching forward via the three term recurrence relation. This process is extremely delicate and is susceptible to numerical perturbations.
Numerical stability is a major concern.
We also develop procedures for testing the mathematical software for its accuracy and efficiency for a range of the parameter b1 and the order of the orthonormal polynomials n. Orthogonal polynomials of these form will have application to approximation theory, spectral methods for solving differential equations, etc.