Infinite products (awaiting review)

Many theta functions have infinite product representations, and this is another important source of modular forms: for instance, the Dedekind eta function $\eta(z)=q^{1/24}\prod_{n\ge1}(1-q^n)$, and $\Delta(z)$$=\eta^{24}(z)=q\prod_{n\ge1}(1-q^n)^{24}$ are fundamental examples (where $q=e^{2\pi iz}$). The modularity of $\eta$ (equivalently of $\Delta$) is not at all clear from the definition as an infinite product; it can be proved by showing that $\eta$ is in fact a theta series using the Jacobi triple product relation, for example.

Since $\eta$ is the basic construction block of infinite products, this implies that many types of infinite products are modular (of course, not products of the type $\prod(1-q^{n^2})$ or $\prod(1-q^n)^n$). As shown by Richard Borcherds, certain products of the type $\prod_{n\ge1}(1-q^n)^{c(n^2)}$ for suitable $c(n)$ (that can in fact be given as Fourier coefficients of other modular forms) are modular.

However, the function $q^c\prod_{n\ge0}(1-q^{5n+1})$ for instance is not modular for any value of $c$, although it closely resembles $\eta$.