Mathematics and Natural Sciences
This paper introduces two families of orthogonal polynomials on the interval (−1,1), with weight function w(x)=1. The first family satisfies the boundary condition p(1)=0, and the second one satisfies the boundary conditions p(-1)=p(1)=0. These boundary conditions arise naturally from PDEs defined on a disk with Dirichlet boundary conditions and the requirement of regularity in Cartesian coordinates. The families of orthogonal polynomials are obtained by orthogonalizing short linear combinations of Legendre polynomials that satisfy the same boundary conditions. Then, the three-term recurrence relations are derived. Finally, it is shown that from these recurrence relations, one can efficiently compute the corresponding recurrences for generalized Jacobi polynomials that satisfy the same boundary conditions.
Lambers, J. V.
(2016). Recurrence Relations for Orthogonal Polynomials for PDEs In Polar and Cylindrical Geometrics. SpringerPlus, 5, 1-29.
Available at: https://aquila.usm.edu/fac_pubs/16745