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.