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Suppose I have a function $f(x)$ with a linear asymptotic behavior for $x\to\infty$: $$f(x)=a_0\,x+b_0+o(1).\tag1$$ Suppose, the last term decays roughly exponentially, so if we take its logarithm, we again get a function with a linear asymptotic behavior: $$\ln(f(x)-(a_0\,x+b_0))=a_1\,x+b_1+o(1),\tag2$$ where $a_1<0$. Repeating the same step again, we get: $$\ln\!\big(\ln(f(x)-(a_0\,x+b_0))-(a_1\,x+b_1)\big)=a_2\,x+b_2+o(1),\tag3$$ and so on.

So we can write the following asymptotic expansion: $$\large{f(x) \sim a_0\,x+b_0+e^{a_1 x+b_1+e^{a_{\small 2}x+b_{\small 2}+e^{...}}}}.\tag4$$

Is there a name for such an expansion? Are there any known expansions of this form? When does it converge?


The motivation for this question is that the fibonorial, that is the product of the Fibonacci numbers $$n!_F=\prod_{m=1}^n F_m\tag5$$ appears to have a similar asymptotic expansion (although so far I do not have a rigorous proof for that): $$n!_F \sim c \cdot 5^{-\frac n 2} \, \phi^{\frac{n\,(n+1)}2} e^{-\frac1{\sqrt5} \frac{(-1)^n}{\phi^{2n+1}} \, \Large e^{-\frac12 \frac{(-1)^n}{\phi^{2n+1}} e^{-\frac1{12} \frac{(-1)^n}{\phi^{2n+1}} e^{-\frac{23}{24} \frac{(-1)^n}{\phi^{2n+1}} e^{\frac{24931}{82800} \frac{(-1)^n}{\phi^{2n+1}} e^{^{...}}}}}}}\tag6$$ where $$c=\prod_{n=1}^\infty\left(1-(-1)^n\phi^{-2n}\right)=\left(-\phi^{-2};\,-\phi^{-2}\right)_\infty\tag7$$ is the Fibonacci factorial constant, and the coefficients on the higher levels (replaced above with the ellipsis) are: $${\small \frac{2869583039}{4128573600},\,\frac{15485246938446577291}{1293723497738290207680},\,\frac{15950805330913927636702617812270117323162487}{1001681391627412794937755717587878089744000},\,...}\tag8$$

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