The general classification problem for stable associative multiplications in complex cobordism theory is considered. It is shown that this problem reduces to the theory of a Hopf algebra S (the Landweber-Novikov algebra) acting on the dual Hopf algebra S* with distinguished 'topologically integral' part A that corresponds to the complex cobordism algebra of a point. We describe the formal group and its logarithm in terms of the algebra representations of S. The notion of one-dimensional representations of a Hopf algebra is introduced, and examples of such representations motivated by well-known topological and algebraic results are given. Divided-difference operators on an integral domain are introduced and studied, and important examples of such operators arising from analysis, representation theory, and non-commutative algebra are described. We pay special attention to operators of division by a non-invertible element of a ring. Constructions of new associative multiplications (not necessarily commutative) are given by using divided-difference operators. As an application, we describe classes of new associative products in complex cobordism theory.