Orthogonal polynomials on a class of planar algebraic curves

Abstract

We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form ym=ϕ(x)$y^{m} = \phi (x)$ in double-struckR2${\mathbb {R}}^2$ where m=1,2$m = 1, 2$ and ϕ is a polynomial of arbitrary degree d, 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$0, \ldots , N$ are computed via the Lanczos algorithm in scriptOfalse(Nd4false)$\mathcal {O}(Nd^4)$ operations.