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 [1] the only permutable sequence of polynomials.

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[1] 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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