Formal calculus, umbral calculus, and basic axiomatics of vertex algebras

The central subject of this thesis is formal calculus together with certain applications to vertex operator algebras and combinatorics. By formal calculus we mean mainly the formal calculus that has been used to describe vertex operator algebras and their modules as well as logarithmic tensor product theory, but we also mean the formal calculus known as umbral calculus. We shall exhibit and develop certain connections between these formal calculi. Among other things we lay out a technique for efficiently proving certain general formal Taylor theorems and we show how to recast much of the classical umbral calculus as stemming from a formal calculus argument that calculates the exponential generating function of the higher derivatives of a composite function. This formal calculus argument is analogous to an important calculation proving the associativity property of lattice vertex operators. We use some of our results to derive combinatorial identities. Finally, we apply other results to study some basic axiomatics of vertex (operator) algebras. In particular, we enhance well known formal calculus approaches to the axioms by introducing a new axiom, "weak skew-associativity," in order to exploit the S3 -symmetric nature of the Jacobi identity axiom. In particular, we use this approach to give a simplified proof that the weak associativity and the Jacobi identity axioms for a module for a vertex algebra are equivalent, an important result in the representation theory of vertex algebras.