Let G(q) be the generating function for partitions such that if k is a part, then it occurs once and k+1 is not a part.
Let H(q) be the generating function for partitions with the same condition plus that 1 is not a part.
These are the left-hand sides of the Rogers-Ramanujan Identities.
I am intrigued by the following unreferenced statement in the wikipedia page:
If q = e2πiτ, then q−1/60G(q) and q11/60H(q) are modular functions of τ.
Do modular forms shed any light on the Rogers-Ramanujan Identities, or is the connection (as far as we know) a curious coincidence?
Is there some class of modular forms whose Fourier series count natural collections of partitions such as those counted by the left-hand sides of the Rogers-Ramanujan Identities? In particular I have in mind the seemingly "non-local" condition that if k is part, then it is distinct and also k+1 is not parts.
In general, how does one tell if a certain generating function (that counts partitions of a certain type, say) is related (by a multiplicative factor like above) to a modular form of some weight for some group (maybe even with some character)?