Characterization of classical type orthogonal polynomials

Abstract:

We characterize the classical type orthogonal polynomials $\{ {P_n}(x)\} _0^\infty$ satisfying a fourth-order differential equation of type \[ \sum \limits _{i = 0}^4 {{\ell _i}(x){y^{(i)}}(x) = {\lambda _n}y(x)} \] where ${\ell _i}(x)$ are polynomials of degree $\leqslant i$ and ${\lambda _n}$ is a constant. They are only the orthogonal polynomials satisfying an orthogonality of the form \[ \langle {\tau _2},P_m^{''}P_n^{''}\rangle + \langle {\tau _1},P_m’P_n’\rangle + \langle {\tau _0},{P_m}{P_n}\rangle = 0\quad {\text {for}}\;m \ne n\] where ${\tau _0},{\tau _1}$, and ${\tau _2}$ are moment functionals.