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

  1. Apr 28, 2013 #1
    What are some examples of functions such that

    [itex] f(nx) = a_{k}f(x)^{k}+...+a_{1}f(x)+a_{0} [/itex]

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

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

    Are these the only possible examples expressible in terms of elementary functions?
     
  2. jcsd
  3. Apr 28, 2013 #2
    [tex] e^{nx} = (e^x)^n [/tex]

    f(x) = ax + b

    log(nx) = log(x) + log n
     
  4. Apr 28, 2013 #3
    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?
     
    Last edited: Apr 28, 2013
  5. Apr 28, 2013 #4

    Office_Shredder

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

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

    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
     
  6. Apr 28, 2013 #5
    Interesting idea for the fractional powers. Unfortunately I don't think polynomials will work for the relationship.

    Suppose
    [itex] f(x) = a_{k}x^{k}+...+a_{0} [/itex] and [itex] g(x) = b_{l}x^{l}+...+b_{0} [/itex]

    where [itex] f(nx) = g(f(x)) [/itex]

    The LHS has degree k, while the RHS has degree k*l, which is greater than k unless l is 1, which corresponds to [itex] f(nx) = b_{1}f(x)+b_{0} [/itex].
     
  7. Apr 29, 2013 #6

    Stephen Tashi

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

    If we define the lowest possible [itex] n [/itex] 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 [itex] G(x) [/itex] in mind then we can ask for the smallest [itex] n [/itex] such that there is some polynomial [itex] P(x) [/itex] of smaller degree than [itex] G(x) [/itex] and [itex] G(x) [/itex] 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.
     
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