q-series

A q-series is a series in a parameter $q$, treated either as a formal power series or as an analytic series, often for $|q|<1$. Its coefficients can encode a sequence, while identities involving infinite products record choices of parts in integer partitions.

For example,where $p(n)$ counts integer partitions of $n$. Expanding each factor as a geometric series chooses the number of parts of size $m$. Formally, each coefficient uses only finitely many factors. This product is a basic link between combinatorics, number theory, and modular forms.