Out of curiosity I'm playing around with the concept of involutory functions. An involutory function (involution) is a function whose composition with itself is the identity function (i.e. ${f \circ f} = id$). In other words, involutions are functions that are their own inverses (i.e. $f = f^{-1}$).
Applying this concept to univariate real polynomials; it seems that all involutory polynomials have degree one(!) and belong to one of these two families:
- $id$: $id(x) = x$ (the identity function, trivially involutory)
- $p_1$: $p_1(x) = {-x + c}$, for some $c \in \mathbb{R}$
One well-known generalization of polynomials are infinite series, so that raises the question: Is there some pretty classification of involutory real infinite series, too?
Surely the examples will be richer than for polynomials.