Spectral approximations on the triangle

In this paper Ive describe a new family of polynomials which are eigenfunctions of a singular Sturm-Liouville problem on the triangle T-2 = {(x, y) : x greater than or equal to 0, y greater than or equal to 0, x + y less than or equal to 1}. The polynomials are shown to be orthogonal over T-2 with respect to a unit a eight function, and may be used in approximations which are exponentially convergent for functions which are infinitely smooth in T-2. The zeros of the polynomials may be used in cubature formulae on T-2.