I have heard that Polylogarithms are very interesting things. The wikipedia page shows a lot of interesting identities. These functions are indeed supposed to have caught the attention of Ramanujan. Moreover they seem to be important in physics for various purposes like Bose-Einstein integrals, which I am not really knowledgeable enough to understand. These are all things I have heard from people after I queried "why polylogarithms are interesting".

So this function

$$Li_s (z) = \sum_{k=1}^\infty \frac{z^k}{k^s}$$

is very interesting and has a lot of useful properties as I can see. Especially for integral values of $s$.

What is bothering me is the following.

Let $f(z) = \sum a_n z^n$ be analytic inside the unit disc. That is, the radius of convergence at least $1$. For simplicity, we assume it is $1$, i.e.,

$$r = \frac{1}{\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}} = 1$$

implying that inside the unit disc, term-by-term differentiation and integration are possible. Since $\left\( a_n \right)^{1/n}$ goes to $1$ as $n$ goes to $\infty$, it is also true that $\left\(na_n \right)^{1/n}$ goes to $1$. Similarly, $\left\(\frac{a_n}{n} \right)^{1/n}$ also goes to $1$. Now, defining

$$f(z,s) = \sum_{n=1}^\infty \frac{a_n z^n}{n^s}$$

it is the case that $f_{2}(z) = f(z,2)$ and in general iterating, $f_n(z) = f(z, n)$ are analytic inside the unit disc. In fact, uniform bounds hold true for compact sets contained in the domain $|z| \leq 1$ and $|s| \leq K$ for arbitrary $K$, and so $f(z,s)$ is analytic in the domain $D^\circ \times \mathbb C$, where $z \in D^\circ$, the open unit disc, and $s \in \mathbb C$.

So any analytic function in the unit disc can be extended just like the logarithm is extended to polylogarithms. But there is a rich theory about polylogarithms, and I haven't heard of any such theories about other functions analytic inside the unit disc.

So, what makes polylogarithm so amenable to an extended theory of polylogarithms yielding so many results? Why is this not giving such an interesting theory for just any other complex functions?