Orthogonal polynomial eigenfunctions of second-order partial differerential equations

In this paper, we show that for several second-order partial differential e quations L[u] =A(x, y)u(xx) + 2B(x,y)u(xy) + C(x,y)u(yy) + D(x,y)u(x) + E(x,y)u(y) = lambda (n)u which have orthogonal polynomial eigenfunctions, these polynomials can be e xpressed as a product of two classical orthogonal polynomials in one variab le. This is important since, otherwise, it is very difficult to explicitly find formulas for these polynomial solutions. From this observation and cha racterization, we are able to produce additional examples of such orthogona l polynomials together with their orthogonality that widens the class found by H. L. Krall and Sheffer in their seminal work in 1967. Moreover, from o ur approach, we can answer some open questions raised by Krall and Sheffer.