**Examples of infinite dimensional involutions**

Edit 2/25/23, as suggested by YCOR below: (Start)

The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford language definition of involution as

- MATHEMATICS: a function, transformation, or operator that is equal to its inverse, i.e., which gives the identity when applied to itself.

In this case the basic involution involves substitution into themselves of infinite sets, say $[P]=(P_1(u_1),P_2(u_1,u_2),P_3(u_1,u_2,u_3),...)$ or $[P]=(1,P_1(u_1),P_2(u_1,u_2),P_3(u_1,u_2,u_3),...)$, of homogeneous polynomials $P_n(u_1,u_2,...u_n)$ multivariate in the arbitrary, independent, commutative indeterminates $u_n$ and of degree no greater than $n$ to obtain the identity set under substitution $[I]=(u_1,u_2,u_3,...)$ or $[I]=(1,u_1,u_2,u_3,...)$, i.e., symbolically,

$[P] = [P]^{-1},$

or, equivalently,

$[P][P] = [P]^2 = [I] ,$

representing the graded substitution operation for $n = 1,2,3,...$

$P_n(P_1(u_1),P_2(u_1,u_2),...,P_n(u_1,...,u_n)) = u_n$.

Below, in the question, I attach additional initial polynomials such as $MI_0(a_0)$ containing only $a_0,b_0,c_0,$ or $d_0$, but these indeterminates can be assigned a fixed value of $1$ and the initial polynomials ignored. In my answer, with a simpler construction, I retain the initial $1$ as a reminder that the initial constants or coefficients of the linear term in the associated o.g.f.s are $1$. (End)

I'm looking for more examples of involutions of the type portrayed below, in which two sets of indeterminates (real or complex) each can be transformed into each other using the same set of rational functions. What properties should I expect such involutions to share? (Except for the designated constraint on one indeterminate of one set, the other indeterminates in that set may be assigned arbitrary values if ultimate convergence of the associated power series is ignored.)

**I) Involution related to multiplicative inversion of exponential generating functions (e.g.f.s):**

Given a convergent Taylor series/e.g.f.

$f(z) = \sum_{n \geq 0} a_n \frac{z^n}{n!} = a_0 \sum_{n \geq 0} \frac{a_n}{a_0} \frac{z^n}{n!}$

with $a_0 \neq 1$, the reciprocal (multiplicative inverse) e.g.f

$f^{-1}(z) = \sum_{n \geq 0} b_n \frac{z^n}{n!}$

is given by an involution represented by the homogeneous rational functions described in OEIS A133314. The first few are

$b_0 = \frac{1}{a_0}[1] = MI_0(a_0)$

$b_1 = \frac{1}{a_0^2}[-a_1] = MI_1(a_0,a_1)$

$b_2 = \frac{1}{a_0^3}[-a_2a_0+2a_1^2 ] = MI_2(a_0,a_1,a_2)$

$b_3 =\frac{1}{a_0^4}[-a_3a_0^2+6a_2a_1a_0-6 a_1^3] = MI_3(a_0,a_1,a_2,a_3)$

$b_4 =\frac{1}{a_0^5}[-a_4a_0^3+8a_3a_1a_0^2+6a_2^2a_0^2-36a_2a_1^2a_0+24a_1^4 ] = MI_4(a_0,a_1,a_2,a_3,a_4) .$

The polynomial in the numerator of $MI_n(a_0,...,a_n)$ is homogeneous of order $n$ while the rational function $MI_n(a_0,...,a_n)$ is homogeneous of order $-1$, i.e., $MI_n(t\cdot a_0,...,t\cdot a_n) = t^{-1} \cdot MI_n(a_0,...,a_n).$ The set of rational functions is an involution in that $b_n = MI_n(a_0,...,a_n)$ and $a_n = MI_n(b_0,...,b_n).$ (With b_0 =a_0=1, the resulting polynomials are the refined Euler characteristics (or signed, refined face polynomials) of the permutahedra. The o.g.f. version is A263633, a refined version of the Pascal triangle.)

**II) Involution related to compositional inversion of e.g.f.s:**

Given the function

$h(z) = \sum_{n \geq 1} c_n \frac{z^n}{n!}$

with $c_1 \neq 0$, the compositional inverse is

$h^{(-1)}(z) = \sum_{n \geq 1} d_n \frac{z^n}{n!}$

with the first few coefficients given in A134685 as

$d_1 = \frac{1}{c_1} [ 1 ] = CI_1(c_1)$

$d_2 = \frac{1}{c_1^3} [ -c_2 ] =CI_2(c_1,c_2) $

$d_3 = \frac{1}{c_1^5} [ 3 c_2^2 - c_1c_3 ]=CI_3(c_1,c_2,c_3) $

$d_4 = \frac{1}{c_1^7} [ -15 c_2^3 + 10 c_1c_2c_3 - c_1^2 c_4 ]=CI_4(c_1,c_2,c_3,c_4) $

$d_5 = \frac{1}{c_1^9} [ 105 c_2^4 - 105 c_1c_2^2 c_3 + 15 c_1^2 c_2c_4 + 10 c_1^2 c_3^2 -c_1^3 c_5 ]=CI_5(c_1,c_2,c_3,c_4,c_5). $

Reducing the indices of the indeterminates by one gives the same partitions as in the first example in the numerators. The polynomial in the numerator of $CI_n(c_1,...,c_n)$ is homogeneous of order $n-1$ while the rational function $CI_n(c_1,...,c_n)$ is homogeneous of order $-n$, i.e., $CI_n(t\cdot c_1,...,t\cdot c_n) = t^{-n} \cdot CI_n(c_1,...,a_n).$ The set of rational functions is an involution in that $d_n = CI_n(c_1,...,c_n)$ and $c_n = CI_n(d_1,...,b_n).$ (The o.g.f version is A133437, the refined Euler characteristics (or signed, refined face polynomials) of the associahedra.)

(Proposition 2.16 on page 15 of "Hopf monoids and generalized permutahedra" by Marcelo Aguiar and Federico Ardila appears relevant.)

**III) Involution related to compositional inversion of Laurent series**

(Added June 10, 2022, from “Differential calculus on the Faber polynomials” by Helene Airault and Abdlilah Bouali based on a result by Schur, with $a_0,b_0$ included. Edited june 25, 2022: Dependency on $a_0$ corrected.)

Given the inverse pair of Laurent series

$f(z) = a_0 \; z + a_1 + \frac{a_2}{z} + \frac{a_3}{z^2} + \cdots$

and

$f^{(-1)}(z) = b_0 \; z + b_1 + \frac{b_2}{z} + \frac{b_3}{z^2} + \cdots,$

the first few equalities for the involution are

$b_0 = \frac{1}{a_0}$

$b_1 = -\frac{a_1}{a_0}$

$b_2 = -a_2$

$b_3 = -(a_1a_2+a_0a_3)$

$b_4 = - (a_1^2a_2 +a_0a_2^2 + + 2 a_0a_1 a_3+a_0^2 a_4)$

$b_5 = -( a_1^3 a_2+ 3 a_0a_1 a_2^2+ 3 a_0a_1^2 a_3+ 3a_0^2 a_2 a_3+3 a_0^2a_1 a_4 + a_0^3 a_5)$.

For example, for the first four grades of involution,

$b_0 = \frac{1}{a_0} = \frac{1}{\frac{1}{b_0}} =b_0$

$b_1 = -\frac{a_1}{a_0} = -\frac{-(b_1/b_0)}{1/b_0} = b_1 $

$b_2 = -a_2 = -(-b_2)= b_2$

$b_3 = -(a_1a_2+a_0a_3) = -[ -(\frac{b_1}{b_0})(-b_2) + \frac{1}{b_0}((-(b_1b_2+b_0b_3))] = b_3 .$

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