On the polynomial representation of generalized Liouville operators

The generalized Liouville operators are given as determinants of square matrices whose entries are higher order derivatives of real valued functions in two variables. The composition of these operators has very interesting properties in connection with polynomials over these operators. Their expansions grow so quickly that it is necessary to build efficient computer algorithms in order to manage them symbolically and to check inclusion properties. We prove that a general polynomial representation of the operator composition cannot exist, and we recognize other polynomial expressions, included in the composition, which group a great quantity of terms.