Division of differential operators, intertwine relations and darboux transformations

Abstract

The problem of a differential operator left- and right division is solved in terms of generalized Bell polynomials for a nonabelian differential unitary ring. The definition of the polynomials is made by means of recurrent relations. The expressions of classic Bell polynomials via a generalized one is given. Conditions of exact factorization lead to intertwine relations and result in linearizable generalized Burgers equation. An alternative proof of the Matveev theorem is given and transformation formulae for the coefficients of differential operator in terms of differential polynomials follow from the intertwine relation.