General orthogonal polynomials and the Jacobi family, which supply the many weighted bases that ApproxFun works with.
Recurrence and zero structure for general measures, followed by weighted expansions under a separate polynomial-density hypothesis.
The Jacobi family: explicit formulas, derivatives, recurrences, orthogonality, exact norms, zeros and completeness.
Ultraspherical, Legendre and Chebyshev polynomials as Jacobi cases, with exact normalising bridges.
Changing Jacobi parameters without changing the represented polynomial: the connection formulas behind basis conversion.
Weighted \(L^2\) and uniform geometric convergence of Jacobi and ultraspherical expansions.
Gaussian quadrature for a general measure and its Golub–Welsch eigenvalue characterisation.