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.



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> $G(q)=\displaystyle\sum\_{n=0}^\infty\frac{q^{n^2}}{(q;q)\_n}=\frac{1}{(q;q^5)\_\infty(q^4;q^5)\_\infty}$
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> $H(q)=\displaystyle\sum\_{n=0}^\infty\frac{q^{n^2+n}}{(q;q)\_n}=\frac{1}{(q^2;q^5)\_\infty(q^3;q^5)\_\infty}$
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I am intrigued by the following unreferenced statement in the wikipedia [page](http://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_identities#Modular_functions%20%22page%22):



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> If q = e2πiτ, then q−1/60G(q) and q11/60H(q) are modular functions of τ.
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1. Do modular forms shed any light on the Rogers-Ramanujan Identities, or is the connection (as far as we know) a curious coincidence?
2. 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.
3. 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)?



What is the relationship between modular forms and the Rogers-Ramanujan identities?
It's hard to compete with Berndt's former student and Berkovich's active collaborator in providing an exhaustive link of references. I can only indicate my own modest [contribution](http://arxiv.org/abs/1001.1571), joint with Ole Warnaar (who is an expert in the business), in which you can find links to further literature as well as discussion of other (not originally expected!) aspects of Rogers-Ramanujan identities.


As for the original question,



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> What is the relationship between Modular Forms and the Rogers-Ramanujan Identities?
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the answer is straightforward: whenever you see Rogers-Ramanujan-type identities, both sides are *modular* forms. It doesn't however work in the opposite direction: there are plenty of modular forms for which an RR-style interpretation isn't known.


