On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev--Jacobi transform

We describe a fast, simple, and stable transform of Chebyshev expansion coefficients to Jacobi expansion coefficients and its inverse based on the numerical evaluation of Jacobi expansions at the Chebyshev–Lobatto points. This is achieved via a decomposition of Hahn’s interior asymptotic formula into a small sum of diagonally scaled discrete sine and cosine transforms and the use of stable recurrence relations. It is known that the Clenshaw–Smith algorithm is not uniformly stable on the entire interval of orthogonality. Therefore, Reinsch’s modification is extended for Jacobi polynomials and employed near the endpoints to improve numerical stability.