1
$\begingroup$

I have been using the function $$f(x)=af(x-1)^2+bf(x-1)+c$$ for a project, but wanted to know if there was a closed form of the equation or a form of the function in relation to $f(0)$ or $f(1)$. If not, is there a way to solve for a, b, and c with 3 or 4 values of $f(x)$? I know that $$f(1)=af(0)^2+bf(0)+c$$ $$f(2)=a^3f(0)^4+2a^2bf(0)^3+2a^2cf(0)^2+ab^2f(0)^2+2abcf(0)+ac^2+abf(0)^2+b^2f(0)+bc+c$$ and so on, but it would be much easier if there was a simpler way of evaluating this equation.

$\endgroup$
6
  • $\begingroup$ just think that f(x) could be $sin(x), e^x, x^2 , e^{sin(x)} you can not evaluate ist if you do not specify f $\endgroup$
    – trula
    May 20 at 20:09
  • $\begingroup$ If $f(k)$ is known for $k=0,1,2,3$ then you have three linear equations $f(1)=af(0)^2+bf(0)+c$, $f(2)=af(1)^2+bf(1)+c$, $f(3)=af(2)^2+bf(2)+c$ from which you can solve for $a,b,c$, assuming there is a solution. $\endgroup$ May 20 at 20:13
  • 1
    $\begingroup$ By $f(x)=(g(x)-b/2)/a$ you can convert it to $g(x)=g(x-1)^2+K$ where $K=-b^2/4+ac+b/2$. I don't think there is a closed form possible for that, maybe in few special cases ... $\endgroup$
    – Sil
    May 20 at 20:15
  • $\begingroup$ What you ask is an equivalent of logistic map. $\endgroup$ May 21 at 9:51
  • $\begingroup$ The OEIS sequence A003095 is a simple example where $\,a=1,b=0,c=1\,$ with lots of references. $\endgroup$
    – Somos
    May 25 at 0:53

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.