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,

*T*_{n}(*x*) = cos (*n *arccos(*x*))

where *T*_{n} is the *n*th 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

*T*_{m}(*T*_{n}(*x*)) = *T*_{mn} (*x*).

because

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

Then the polynomials *T*_{m} and *T*_{n} must be permutable because

*T*_{m}(*T*_{n}(*x*)) = *T*_{mn} (*x*) = *T*_{n}(*T*_{m}(*x*))

for all *x*.

There’s one more family of polynomials that are permutable, and that’s the power polynomials *x*^{k}. They are trivially permutable because

(*x*^{m})^{n} = (*x*^{n})^{m}.

It turns out that the Chebyshev polynomials and the power polynomials are essentially [1] the only permutable sequence of polynomials.

## Related posts

[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.

Interesting. Thanks for sharing!

Chebyshev polynomial is a fascinating objects indeed.