In learning about asymptotic expansions of functions, I've encountered several problems where a particular pattern of powers is coming into play, and I'm finding functions that I can readily show to be asymptotically equivalent, having those powers alone in their asymptotic expansions out to any degree of expansion. Let me give a few examples to clarify this before I ask my question.
Example #1: The positive integers are given as powers. The function $\exp(\epsilon)-1$ readily has $\sum_{k=1}^n\frac{1}{n!}\epsilon^n$ as an $n$-term asymptotic expansion for all $n\geq 1$. I could also have used $\frac{\epsilon}{1-\epsilon}=\frac{1}{1-\epsilon}-1$, which has $\sum_{k=1}^n\epsilon^n$ as an $n$-term asymptotic expansion.
Example #2: The positive odd integers are given as powers. The function $\sin(\epsilon)$ readily has $\sum_{k=0}^{n-1}\frac{(-1)^k}{(2k+1)!}\epsilon^{2k+1}$ as an $n$-term asymptotic expansion for all $n\geq 1$.
Example #3: The non-negative even integers are given as powers. $\cos(\epsilon)-1$ readily has $\sum_{k=1}^n\frac{(-1)^k}{(2k)!}\epsilon^{2k}$ as an $n$-term asymptotic expansion for all $n\geq 1$. I could also have used $\frac{\epsilon^2}{1-\epsilon^2}=\frac{1}{1-\epsilon^2}-1$, which has $\sum_{k=1}^n\epsilon^{2n}$ as an $n$-term asymptotic expansion.
In the above examples, all I had to do was think about some functions with that particular pattern to the powers appearing in their MacLaurin series, and simply drop the tails. For the next situation, though, I couldn't think of any such function. I wonder if anyone has any ideas? In particular, I'd be interested if someone knows of a function $f$ with the following specific properties:
(i) $f(0)=0$ and $f$ is non-$0$ on some punctured neighborhood of $0$.
(ii) The powers given are the integers of the form $k^2+1$, $k\geq 0$--that is, I'd like to have $f(t)=\sum_{k=0}^\infty a_kt^{k^2+1}$ for some non-$0$ constants $a_k$ in some neighborhood of $t=0$.
(iii) $f$ is constructed via composition and basic arithmetic operations (PEMDAS) from exponential, logarithmic, trig, inverse trig, and polynomial functions as in the examples above (this is what I mean by "nice" functions).
Does such a function $f$ exist (that anyone know of)? If not, why not? If we remove requirement (iii), is there any big-name "not-so-nice" function satisfying the other two properties?
EDIT: In (iii), I'll allow $n$th roots as well, so long as it doesn't break things (e.g.: keep us from even having a MacLaurin series).