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 \(\mathbb{R}^2\) where \(m = 1, 2\) and \(\phi\) is a polynomial of arbitrary degree \(d\), in terms of univariate semiclassical OPs. We compute connection coeffeicients 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 \(0, \ldots, N\) are computed via the Lanczos algorithm in \(O(Nd^4)\) operations.