Orthogonal polynomials on a class of planar algebraic curves

We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form y(m) =phi(x) in R-2 where m= 1, 2 and phi is a polynomial of arbitrary degree.., in terms of univariate semiclassical OPs. We compute connection coefficients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrix is shown to be banded and the connection coefficients and Jacobi matrices for OPs of degree 0,.,. N are computed via the Lanczos algorithm in O (Nd-4) operations.