I want to find a smooth function $F$ satisfies $\ln F(x+1)=a F(x),\ x\in[0,2]$ and $F(0)=1$
I didn't prove the existance of the function but I think it exists.
I can easily get $F(1)=e^a$ and $F(2)=e^{ae^a}$
I found it hard to evaluate it. So I only want to find the coefficients of Taylor series of $F$.
So I let $$F(x)=\sum_{k\geq0}c_kx^k\text{ (suppose it is also smooth in [-2,0])}$$ And I can get$$\ln F(x+1)\\ =\ln(1+\sum_{k\geq1}{\frac{c_k}{k!}(x+1)^k})\\ =\sum_{n\geq1}{\sum_{k\geq1}{(-1)^{n+1}(\sum_{m\geq0}\frac{c_k}{k!}C_k^mx^m)^n}}$$ I want to compare the coefficients of $x$ to get the value of $c_k$ but it looks like impossible for me.
Then, I had a thought: transfer the equation into $$F(x)=\exp(aF(x-1))\text{ or}\\\ln F(x)=aF(x-1)$$ But all of these contain $\ln F(...)$ or $\exp F(...)$
So, I have to evaluate formulas like $(a+bx+cx^2+...)^n$.
I tried using computer to find the coefficients but there are infinite terms of $c_0$. I can't treat them well.
On the other hand, I found it's increment speed is very fast, faster than $e^x$, $x!$, $x^{x^x}$ and many other functions. I don't think it is an easy question. What can I do?
Edit: I have no idea about $F(x),\ x\in[0,1)$, but I think the smoothness of $F$ can make it unique.

  • $\begingroup$ a) it is not an easy task infact b)the initial conditions shall be established for all $F(x)\quad |0 \le x <1$ or other equivalent bounds (similar to Gamma function) c) Taylor series does not look to be appropriate. $\endgroup$ – G Cab Feb 22 '18 at 14:31
  • $\begingroup$ First define a function $F$ on $[0,\epsilon)$ and $(1-\epsilon,1]$ such that $F(0) = 1$, $F(1) = e^a$ and $F^{(n)}(0) = F^{(n)}(1) = 0$ for all $n > 0$. Next extend the definition to $[0,1]$ by joining them with a smooth function and finally extend definition to $[1,2]$ using the functional equation. e.g. $$F(x) = \begin{cases} 1,& x = 0\\ \frac{e^a+1}{2} + \frac{e^a-1}{2}\tanh(\tan(\frac{\pi}{2}(2x-1)),& x \in (0,1)\\ e^a, & x = 1\\ e^{aF(x-1)},& x \in (1,2] \end{cases}$$ $\endgroup$ – achille hui Feb 22 '18 at 14:45

This problem is called "tetration". Discussion can be seen on the Wikipedia page on Tetration

A paper about it can be found at http://myweb.astate.edu/wpaulsen/tetration2.pdf


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