One-Dimensional Orthogonal Polynomials

In the following sections we use make considerable use of the Jacobi polynomials. These are defined as the polynomial solutions of the following Sturm-Liouville problem:

They have the following orthogonality:

$$\int^{1}_{-1} (1-x)^{\alpha} (1+x)^{\beta} P^{\alpha,\beta}_n(x) P^{\alpha,\beta}_m(x) dx = 0 \textstyle\hspace{1cm}n\neq m$$

As the following sections were just being completed I found that work along exactly the same lines had just been submitted to the SIAM Journal on Numerical Analysis by Wingate and Taylor [28]. I include the work here for completeness noting that the results have already been independently obtained and presented there, except for the results based on prisms.