New perspectives on exponentiated derivations, the formal Taylor theorem, and Fa\`a di Bruno's formula

TL;DR

This paper explores a new perspective on formal calculus used in vertex algebras, emphasizing the role of formal translation operators and their extensions, connecting to Faà di Bruno's formula and umbral calculus.

Contribution

It introduces a viewpoint that simplifies extensions of formal calculus, facilitating the inclusion of logarithmic and iterated-logarithm objects, and links to classical combinatorial formulas.

Findings

Formal translation operator approach simplifies extensions of formal calculus.

Connections established between formal calculus, Faà di Bruno's formula, and umbral calculus.

Extensions include logarithmic and iterated-logarithm objects in formal calculus.

Abstract

We discuss certain aspects of the formal calculus used to describe vertex algebras. In the standard literature on formal calculus, the expression $(x+y)^{n}$, where $n$ is not necessarily a nonnegative integer, is defined as the formal Taylor series given by the binomial series in nonnegative powers of the second-listed variable (namely, $y$). We present a viewpoint that for some purposes of generalization of the formal calculus including and beyond "logarithmic formal calculus", it seems useful, using the formal Taylor theorem as a guide, to instead take as the definition of $(x+y)^{n}$ the formal series which is the result of acting on $x^{n}$ by a formal translation operator, a certain exponentiated derivation. These differing approaches are equivalent, and in the standard generality of formal calculus or logarithmic formal calculus there is no reason to prefer one approach over the other. However, using this second point of view, we may more easily, and in fact do, consider extensions in two directions, sometimes in conjunction. The first extension is to replace $x^{n}$ by more general objects such as the formal variable $\log x$, which appears in the logarithmic formal calculus, and also, more interestingly, by iterated-logarithm expressions. The second extension is to replace the formal translation operator by a more general formal change of variable operator. In addition, we note some of the combinatorics underlying the formal calculus which we treat, and we end by briefly mentioning a connection to Fa\`a di Bruno's classical formula for the higher derivatives of a composite function and the classical umbral calculus. Many of these results are extracted from more extensive papers \cite{R1} and \cite{R2}, to appear.