Functions for which f(nx) is a polynomial of f(x).

[Boorglar]

What are some examples of functions such that

$f(nx) = a_{k}f(x)^{k}+...+a_{1}f(x)+a_{0}$

for some integers n, k, and integer coefficients in the polynomial?

The only example I can think of is cos(x), for which $\cos(2x) = 2\cos(x)^{2}-1$ and there are similar relations for n = 3, 4, etc.

Are these the only possible examples expressible in terms of elementary functions?

 

[willem2]

$$e^{nx} = (e^x)^n$$

f(x) = ax + b

log(nx) = log(x) + log n

 

[Boorglar]

Ah yes I didn't think of those.

Are there any more "exotic" examples?
I mean, for which the polynomial involved has more than 2 terms, say?

 

[Office_Shredder]

Nicely factored polynomials let you write the opposite of polynomials (instead of integer terms, 1/k terms)

$$f(x) = (x+1)^2$$
Then
$$f(nx) = (nx+1)^2 = n^2(x+1/n)^2 = n^2 (x+1+(1/n-1))^2$$
$$= n^2 ( (x+1)^2 + 2(x+1)(1/n-1) + (1/n-1)^2)$$
$$= n^2(f(x)+ (1/n-1) f(x)^{1/2} + (1/n-1)^2)$$

I can't figure out a way to jigger it (make it negative degree terms, fractional terms etc) to make it a true polynomial but maybe someone else can see how to do it

 

[Boorglar]

Interesting idea for the fractional powers. Unfortunately I don't think polynomials will work for the relationship.

Suppose
$f(x) = a_{k}x^{k}+...+a_{0}$ and $g(x) = b_{l}x^{l}+...+b_{0}$

where $f(nx) = g(f(x))$

The LHS has degree k, while the RHS has degree k*l, which is greater than k unless l is 1, which corresponds to $f(nx) = b_{1}f(x)+b_{0}$.

 

[Stephen Tashi]

Are there any more "exotic" examples?
I mean, for which the polynomial involved has more than 2 terms, say?

If $f(nx) = P(f(x))$ then $f(n^2x) = f(n(nx)) = P(f(nx)) = P(P(f(x))$. If the degree of the polynomial $P(x) > 1$ you can build a sequence of higher degree polynomials by changing $n$ to powers of itself.

If we define the lowest possible $n$ to be a "non-trivial" solution, the question can be rephrased to ask for examples where a non-trivial solution is a polynomial of degree greater than 2.

If we have a particular polynomial $G(x)$ in mind then we can ask for the smallest $n$ such that there is some polynomial $P(x)$ of smaller degree than $G(x)$ and $G(x)$ is a member of the sequence of polynomials that are built-up in the above fashion. Thats a question for a good algebraist and I'm not sure how the answer bears on the original question - I'll use the excuse that it past 3AM in my time zone.