Two polynomials P(x) and Q(x) commute whenever P(Q(x)) = Q(P(x)).

Some polynomials commute, and others don't. The set of powers \(\{x^1, x^2, x^3, \ldots, \}\) is an example of a chain—a list of polynomials, one of each degree, which all commute with each other. Are there any other chains?

Any linear polynomial \(\lambda(x)=ax+b\), (where a is nonzero), has an inverse \(\lambda^{-1}(y)=(y-b)/a\). And if \(\{P_i(x)\}\) is a chain, then \(\{\lambda^{-1}\circ P_i \circ \lambda\}\) is also a chain. So every chain is part of a family of similar polynomials. Up to similarity, there is only one other polynomial chain: the Chebyshev polynomials \(T_n(x) \equiv \cos(n\cdot \cos^{-1}(x))\).

Why? You can prove that any quadratic polynomials commutes with at most one polynomial of each degree (so the quadratic determines the whole chain), and each quadratic is similar to a quadratic of the form \(x^2+c\). In chains, the commuting constraint forces \(c(c+2)=0\). When c=0, you get the power polynomials. When c=-2, you get the Chebyshev polynomials.  

Reference 1, Reference 2.