# References to Information on Alternative Series Representations

I've come across a problem with the size of coefficients of a series expansion. Here's an example of an expression that I expand into a power series:

$\displaystyle\frac{1-e^{i3n\cdot t}}{1-e^{i\cdot n\cdot t}}$ where $n \in \mathbb{N}$ and $t \in \mathbb{R}$

The problem is that I'd like to work with small coefficient values, but the values increase exponentially with $n$.

One question that seems natural to ask in this case is, "Can we find another series with smaller coefficients?". Let me explain a little. I'm fiddling with an integration technique for specific forms of integrals, and I use the series expansions to help with the integration. I can fairly easily work with more terms, but it's hard to work with larger values of coefficients. This may seem a bit contrary to many mathematicians' experiences.

What I'm getting at is that I'd like to find alternatives to power series expansions. In other words, is there literature on series expansions other than power series expansions? I've seen series like Dirchlet series and whatnot, but I'm specifically interested in finding series expansions for functions of exponentials. And, of course, I'm hoping that I can find ones whose terms are smaller than the corresponding power series expansions.

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I'm not entirely sure how helpful this suggestion (this was supposed to be a comment, but it got very long) will ultimately be, but you might be interested in knowing that there exists a generalization of the usual Taylor series, called the (Lagrange-)Bürmann series.

Briefly, given a function $f(x)$ to be expanded, a "basis function" $g(x)$, an expansion point $a$, and the assumption that $g^{\prime}(x)$ is nonzero over some interval containing $a$, such that $g(x)-g(a)$ is monotonic as $x$ increases over said interval, the Bürmann series of $f(x)$ with basis function $g(x)$ and expansion point $x=a$ reads as

$$f(x)=f(a)+\sum_{k=1}^{\infty} \frac{\beta_k(a)}{k!}(g(x)-g(a))^k$$

where the $\beta_k(x)$ satisfy the recursion

$$\beta_0(x)=f(x),\qquad \beta_k(x)=\frac1{g^{\prime}(x)}\frac{\mathrm d}{\mathrm dx}\beta_{k-1}(x)$$

The series may or may not converge, of course, so the analysis of convergence (which highly depends on the nature of your $f(x)$ and $g(x)$) is still your burden.

It's not terribly popular, probably because it's difficult to choose $g(x)$ such that your expansion of $f(x)$ is meaningful, but it's worth a shot.

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This is worthwhile of exploration, so +1 - but I probably won't accept this as an answer. Good ideas are welcome! – Matt Groff Nov 23 '10 at 3:46
I'm not expecting this to be an accepted answer, FWIW. ;) I'm just throwing it out and hoping somebody smarter than me figures out how to make something special from this. – J. M. Nov 23 '10 at 3:50
Do you have a reference to a resource (such as a text or url) that goes into more detail on these series? (Hopefully something more than Wikipedia) – Matt Groff Nov 23 '10 at 3:58
@Matt: It's from my (yellowed) notes, but I'll try to look for the book(s) where I got this from (but as I seem to recall, Whittaker and Watson mention this in passing in their book). FWIW, a search on Google turns up precious few... – J. M. Nov 23 '10 at 4:04