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.

$G(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2}}{(q;q)_n}=\frac{1}{(q;q^5)_\infty(q^4;q^5)_\infty}$

$H(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2+n}}{(q;q)_n}=\frac{1}{(q^2;q^5)_\infty(q^3;q^5)_\infty}$

I am intrigued by the following unreferenced statement in the wikipedia page:

If q = e

^{2πiτ}, then q^{−1/60}G(q) and q^{11/60}H(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)?

Introduction to Modular Forms). Alas, I know no direct way of showing that the left sides are modular forms, short of proving the RR identities. I would appreciate any references to such a proof. – Robin Chapman Jun 23 '10 at 9:27