Let $A$ be a function from $[0,1]$ to $[0,1]$. $A$ is an *involution* if $A(A(x))=x$ for all $x\in[0,1]$.

Which involutions $A$ exist such that $A(x)=\sum_{k=0}^\infty a_k x^k$ with $a_0=1$ and $a_k\leq0$ for all $k\geq1$?

The condition on the form of $A$ is equivalent to having that $A(x)=1-G(x)$ where $G(x)=\sum_{k=1}^\infty p_k x^k$ is the probability generating function of a probability distribution on the positive integers.

Here are three families of solutions:

$A(x)=\frac{1-x}{1-(1-p)x}$ for $p\in(0,1)$. Then all the coefficients $a_k$ are non-zero, and in fact $G=1-A$ is the probability generating function of the geometric distribution with parameter $p$ (taking values in $\{1,2,3,\dots\}$).

$A(x)=1-\left[1-(1-x)^{1/n}\right]^n$ for $n=1,2,3,\dots$. Then the non-zero coefficients are $a_0$ and $a_n, a_{n+1}, a_{n+2},\dots$.

$A(x)=(1-x^n)^{1/n}$ for $n=1,2,3,\dots$. Then the non-zero coefficients are $a_0, a_n, a_{2n}, a_{3n}, \dots$.

Are there any more solutions? Are there lots more solutions?!

(Note that if you take non-integer $n$ in 2. or 3., you still get involutions but they no longer have power-series expansions as desired.)

[My motivation is a fixed point equation for probability distributions which comes from considering minimax functions (of the sort that arise for example in the analysis of combinatorial games) on random trees. In particular, suppose $G=1-A$ is a pgf, suppose $M$ and $N_1, N_2, N_3, \dots$ are i.i.d. random variables with the distribution whose pgf is $G$, and suppose $Y_{ij}, i,j\in\mathbb{N}$ are i.i.d. real-valued random variables with any common distribution. Let $$ X=\max_{1\leq i\leq M}\min_{1\leq j\leq N_i} Y_{ij}. $$

The property that $A$ is an involution is equivalent to the property that for any common distribution of the $Y_{ij}$, the distribution of $X$ is the same as that of each of the $Y_{ij}$.]