Orthogonal Polynomial Eigenfunctions of Second-order Partial Differerential Equations

In this paper, we show that for several second-order partial differential equations L[u] = A(x, y)uxx + 2B(x, y)uxy + C(x, y)uyy +D(x, y)ux + E(x, y)uy = λnu which have orthogonal polynomial eigenfunctions, these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable. This is important since, otherwise, it is very difficult to explicitly find formulas for these polynomial solutions. From this observation and characterization, we are able to produce additional examples of such orthogonal polynomials together with their orthogonality that widens the class found by H. L. Krall and Sheffer in their seminal work in 1967. Moreover, from our approach, we can answer some open questions raised by Krall and Sheffer.