# Permutable polynomials

Two polynomials p(x) and q(x) are said to be permutable if p(q(x)) = q(p(x)) for all x.

It’s not hard to see that Chebyshev polynomials are permutable.

First, Tn(x) = cos (n arccos(x)) where Tn is the nth Chebyshev polyomial.

You can take this as a definition, or if you prefer another approach to defining the Chebyshev polynomials it’s a theorem.

Then it’s easy to show that Tm(Tn(x)) = Tmn (x).

because cos(m arccos(cos(n arccos(x)))) = cos(mn arccos(x)).

Then the polynomials Tm and Tn must be permutable because Tm(Tn(x)) = Tmn (x) = Tn(Tm(x)) for all x.

There’s one more family of polynomials that are permutable, and that’s the power polynomials xk.

They are trivially permutable because (xm)n = (xn)m.

It turns out that the Chebyshev polynomials and the power polynomials are essentially  the only permutable sequence of polynomials.

Related posts Generalization of power polynomials Product of Chebyshev polynomials The Brothers Markov  Here’s what “essentially” means.

A set of polynomials, at least one of each positive degree, that all permute with each other is called a chain.

Two polynomials p and q are similar if there is an affine polynomial λ(x) = ax + b such that p(x) = λ-1( q( λ(x) ) ).

Then any permutable chain is similar to either the power polynomials or the Chebyshev polynomials.

For a proof, see Chebyshev Polynomials by Theodore Rivlin.

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