A construction of real weight functions for certain orthogonal polynomials in two variables

H.L. Krall and I.M. Sheffer considered the problem of classifying certain second-order partial differential equations having an algebraically complete, weak orthogonal bivariate polynomial system of solutions. Two of the equations that they considered are(x2+y)uxx+2xyuxy+y2uyy+gxux+g(y−1)uy=λu,(x2+y)uxx+2xyuxy+y2uyy+gxux+g(y−1)uy=λu, andx2uxx+2xyuxy+(y2−y)uyy+g(x−1)ux+g(y−γ)uy=λu.x2uxx+2xyuxy+(y2−y)uyy+g(x−1)ux+g(y−γ)uy=λu. Even though they showed that these equations have a sequence of weak orthogonal polynomial solutions, they were unable to show that these polynomials were, in fact, orthogonal. The orthogonality of these two polynomial sequences was recently established by Kwon, Littlejohn, and Lee solving an open problem from 1967.In this paper, we construct explicit weight functions for these two orthogonal polynomial sequences, using a method first developed by Littlejohn and then further developed by Han, Kim, and Kwon. Moreover, two additional partial differential equations were found by Kwon, Littlejohn, and Lee that have sequences of orthogonal polynomial solutions. These equations are given by(x2−x)uxx+2xyuxy+y2uyy+(dx+e)ux+(dy+h)uy=λu,(x2−x)uxx+2xyuxy+y2uyy+(dx+e)ux+(dy+h)uy=λu,xuxx+2yuxy+(dx+e)ux+(dy+h)uy=λu.xuxx+2yuxy+(dx+e)ux+(dy+h)uy=λu. In each of these examples, we also produce explicit orthogonalizing weight functions.