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 formequation(∗)Auxx+2Buxy+Cuyy+Dux+Euy=λu,Auxx+2Buxy+Cuyy+Dux+Euy=λu, and are orthogonal relative to a symmetric bilinear form defined byφ(p,q)=〈σ,pq〉+〈τ,pxqx〉,φ(p,q)=〈σ,pq〉+〈τ,pxqx〉, where A,…,EA,…,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(⋅,⋅)φ(⋅,⋅). Also examples are provided.