Abstract :
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,
and
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, Littlejohnand 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,
xuxx +2yuxy +(dx +e)ux +(dy + h)uy = λu.
In each of these examples, we also produce explicit orthogonalizing weight functions.
© 2005 Elsevier Inc. All rights reserved
Keywords :
Second-order partial differential equation , Weak orthogonalpolynomials , Polynomial killers , Bivariate orthogonal polynomials , Weight functions
