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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).

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    Another term to search for (instead of "theta functions") is "q-functions". I have been recently reading (and re-reading and re-re-reading ...) Bruce Berndt's "Number Theory in the Spirit of Ramanujan" and been amazed at the types of results there, which are based on q-series. I also have become aware of how extraordinary a mathematician Jacobi was.2012-09-13

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Expanding a bit on GEdgar's comment:

The exponential, etc., functions have power series in which the exponents belong to an arithmetic progression, or a union of several arithmetic progressions, or a union of several arithmetic progressions and a finite set, but in any event a set of positive density (in the integers). Combining such functions won't alter that property.

Functions like $\sum t^{k^2}$ go by the name of theta functions, and there is a vast literature about them. They are analytic, so their zeros must be isolated, which would seem to take care of property (i). The Wikipedia essay would be a starting place for learning about their properties.

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    Excellent, thank you!2012-09-13