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.

Related posts Generalization of power polynomials Product of Chebyshev polynomials The Brothers Markov [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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