TITLE: Is every series that matters a Taylor series?
QUESTION [7 upvotes]: Fourier series are Taylor series in complex z, so I'm wondering if there is any kind of series that represents analytic almost everywhere functions that is foundationally distinct from Taylor series?
Or is every series that matters for analytic functions a Taylor series?

REPLY [2 votes]: There are also the series $$\sum_{(a,b) \in \mathbb{Z}^2 \setminus\{(0,0)\}} \frac{1}{(a\tau + b)^{2k}},$$ where $k \geq 2$.  Though these converge on the open set Im$(\tau) > 0$.  They are important in the theories of modular forms and elliptic functions.  They satisfy infinitely many
symmetries !