L.J. Burchnall (1892-1975) proved a converse to the linearization formula for Hermite polynomials in 1941. In doing so, he calculated powers of a differential operator in terms of Hermite polynomials and (standard) derivatives. We show that this approach generalizes to all families of orthogonal polynomials in the Askey-scheme and its $q$- analog. We discuss several applications of this approach. One application lies in expansion formulas, and a related application is related to the Lax pair formalism for the Toda lattice.
It is possible to extend this approach to some families of matrix-valued orthogonal polynomials and
a corresponding non-abelian Toda lattice. The lecture reports on joint work with Mourad Ismail (University of Central Florida, USA) and Pablo Román (UNC).