A recursion identity for formal iterated logarithms and iterated exponentials

TL;DR

This paper establishes a recursive identity for formal iterated logarithms and exponentials, extending logarithmic formal calculus and revealing connections to combinatorial identities and vertex operator algebra theory.

Contribution

It introduces a new recursive identity for formal iterated logarithms and exponentials, expanding the mathematical framework used in logarithmic tensor category theory.

Findings

Derived a recursive identity involving formal iterated logarithms and exponentials.

Connected the identity to classical combinatorial identities and Stirling numbers.

Applied the identity to formal series expansions related to vertex operator algebras.

Abstract

We prove a recursive identity involving formal iterated logarithms and formal iterated exponentials. These iterated logarithms and exponentials appear in a natural extension of the logarithmic formal calculus used in the study of logarithmic intertwining operators and logarithmic tensor category theory for modules for a vertex operator algebra. This extension has a variety of interesting arithmetic properties. We develop one such result here, the aforementioned recursive identity. We have applied this identity elsewhere to certain formal series expansions related to a general formal Taylor theorem and these series expansions in turn yield a sequence of combinatorial identities which have as special cases certain classical combinatorial identities involving (separately) the Stirling numbers of the first and second kinds.