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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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are the $a_k$ given? or can they be anything? –  Robert Mastragostino Sep 13 '12 at 0:41
    
They can be anything. –  Cameron Buie Sep 13 '12 at 0:46
    
For exponents a quadratic polynomial (like $k^2+1$) I suggest a theta function. The "elementary" functions you request, however, do not have exponents like this. –  GEdgar Sep 13 '12 at 0:47
    
I've heard of zeta functions, GEdgar, but not theta functions. If you'd like to use them to answer, I'll upvote. If it goes a while without anyone coming up with a "nice" function and nobody proves that no such function can exist, I'll accept it, too, even though it isn't quite what I'm hoping for. –  Cameron Buie Sep 13 '12 at 0:52
    
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. –  marty cohen Sep 13 '12 at 1:15
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1 Answer

up vote 1 down vote accepted

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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Thanks, Gerry! I'll look into it. Should I infer that I have to use theta functions to get both property (i) and property (ii)? Or is there some (faint) hope that I might somehow end up with all 3 properties? –  Cameron Buie Sep 13 '12 at 3:28
    
You don't have to use theta functions to get (i) and (ii); you could use any constants $a_k$ so long as they don't grow so fast that your radius of convergence is zero. But if you want to use functions that people know stuff about, as opposed to doing all the work yourself, then theta functions seem like the way to go. As for all three properties, what I've written isn't a rigorous proof of anything, maybe there's a way around it - but I'd try to prove it before I'd try to find a counterexample. –  Gerry Myerson Sep 13 '12 at 6:36
    
Excellent, thank you! –  Cameron Buie Sep 13 '12 at 11:28
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