Extensions of Umbral Calculus Ii: Double Delta Operators, Leibniz Extensions and Hattori-stong Theorems

We continue our programme of extending the Roman-Rota umbral calculus to the setting of delta operators over a graded ring E∗ with a view to applications in algebraic topology and the theory of formal group laws. We concentrate on the situation where E∗ is free of additive torsion, in which context the central issues are number-theoretic questions of divisibility. We study polynomial algebras which admit the action of two delta operators linked by an invertible power series, and make related constructions motivated by the Hattori-Stong theorem of algebraic topology. Our treatment is couched purely in terms of the umbral calculus, but inspires novel topological applications. In particular we obtain a generalised form of the Hattori-Stong theorem.