Sobolev orthogonal polynomials in two variables and second order partial differential equations

We consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form Auxx +2Buxy + Cuyy + Dux +Euy = λu, (∗) and are orthogonal relative to a symmetric bilinear form defined by ϕ(p, q) = σ,pq + τ,pxqx , where A, . . . , E are polynomials in x and y, λ is an eigenvalue parameter, σ and τ are linear functionals on polynomials. We find a condition for the partial differential equation (∗) to have polynomial solutions which are orthogonal relative to a symmetric bilinear form ϕ(·,·). Also examples are provided. © 2005 Elsevier Inc. All rights reserved