Let be the space of all polynomials with real coefficients. A linear functional is said to be semiclassical if there exists two polynomials and such that the following distributional Pearson equation holds
where linear functionals and are defined as
A semiclassical linear functional is said to be of class if
where is the set of all possible pairs of polynomials from Pearson equation (1). Let be two quasi-definite linear functionals on and let be corresponding sequences of monic orthogonal polynomials, i.e.,
where and is Kronecker delta. The following problem is completely solved: “Describe the pairs of quasi-definite linear functionals for which the differential relation
holds for all positive integer with .” The authors prove that at least one of the functionals is semiclassical of class at most 1 if the differential relation (2) holds, and this fact gives a clue to the structure of pairs .