On the power series of involutory functions

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.