Take the tour ×  Dear all, this is perhaps a bit a vague
question, but some references would already be very helpfull.
So let $G$ be a finitely generated group and choose some finite
set of generators. This allows to define the length of a group
element as usual, by the minimal number of generators needed to
write it as a product of these generators. Using this length you
can define all sort of analytic functions by replacing the "n"
in various summation formulas by the $\mathrm{Length}(g)$ and
the summation is now over the group elements $g$. To have a
concrete example in mind, the "exponential series" is now e.g.
\begin{equation} \mathrm{lexp}(z) = \sum_{g \in G}
\frac{z^{\mathrm{Length}(g)}}{\mathrm{Length}(g)!}.
\end{equation} Since for a given length there are at most
exponentially many group elements, the length-exponential
function is entire. For $G = \mathbb{Z}$ and $1$ as generator
this reproduces the usual exponential series up to a factor $2$
as negative and positive $n \in \mathbb{Z}$ contribute