On the power series of involutory functions

TL;DR

This paper demonstrates that coefficients of involutory functions expressed as power series can be formulated using multivariable Lah polynomials, revealing their structure as conjugates of negative identity.

Contribution

It introduces a novel expression of involutory function coefficients via multivariable Lah polynomials and provides a constructive proof of their conjugacy to negative identity.

Findings

Coefficients of involutory functions relate to multivariable Lah polynomials.

Involutory functions can be viewed as conjugates of negative identity.

Constructive proof of the main relation is provided.

Abstract

It is shown that the coefficients of any involutory function $f$ represented as a power series can be expressed in terms of multivariable Lah polynomials. This result is based on the fact that any such $f~(\neq\text{identity})$ can be regarded as a (compositional) conjugate of negative identity. Moreover, a constructive proof of this statement is given.