# What is the relationship between modular forms and the Rogers-Ramanujan identities?

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 = e2πiτ, then q−1/60G(q) and q11/60H(q) are modular functions of τ.

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)?

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It's fairly straightforward that the right sides of these formulas are modular forms (see the definition of Siegel forms in the last chapter of Lang's 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
@Robin: Yes, you are right, in the classical RR identities and the majority of generalisations, the right-hand sides are obviously modular as products of Dedekind's etas. However, more general RR identities (e.g., ones which appear in the preprint citedd in my response) the right-hand sides are different sums for which one can still show the modularity as they are character sums for certain Virasoro algebras (but this is very sophisticated!). For the left-hand sides there is no clear way to recognise the modularity; I would guess this is related to troubles with bijective proofs... –  Wadim Zudilin Jun 23 '10 at 11:54
@Vladimir: Your additional questions are hard and, in a certain sense, not well posed. For example, there are quite different opinions of whether the relationship of RRs with the modular world is coincidental. My own philosophy is that modular forms are a natural source of many sophisticated identities, as modular forms are a unique object: they satisfy two different types of functional equations. I would really suggest you to answer your items 1-3 yourself after browsing the suggested references. –  Wadim Zudilin Jun 23 '10 at 12:05

## 3 Answers

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, 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,

What is the relationship between Modular Forms and the Rogers-Ramanujan Identities?

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.

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Hi Wadim. I was going to call your attention to this question if you made no type of response, as I was pretty sure you knew all about it. I did not realize you had published in this exact topic. –  Will Jagy Jun 23 '10 at 1:51

At the root, these identities arise because there exist theta function identities (e.g. Jacobi triple product) which connect infinite series to infinite products. The infinite products have partition-theoretic interpretation as number of partitions of certain type mod k - etc while the q-series (generating functions) are also modular functions, which satisfy modular equations between moduli. By virtue of this, two types of partitions get connected into a partition identity. The Bailey lemma also comes up in this context.

David Bressoud in his book Analytic and combinatorial generalizations of the Rogers-Ramanujan identities explains that Rogers-Ramanujan identities can be stated combinatorially (set bijection) or analytically (using the function theory of Riemann surfaces) and each approach has generalizations. The analytic statement was discovered by Rogers, Ramanujan and Schur and the combinatorial statement was discovered by MacMahon and Schur.

Generalizations have been proved - see Gordon-Gollnitz identities and Andrews-Gordon identity

In addition to the links given by Will Jagy, a couple of papers listed below by Bruce Berndt discuss how modular equations of various degree are linked to certain types of partitions.

http://www.math.uiuc.edu/~berndt/publications.html

1. Partition identities and Ramanujan's modular equations (with N. D. Baruah), J. Comb. Thy. (A) 114 (2007), 1024-1045 (pdf).
2. Partition identities arising from theta function identities (with N. D. Baruah), Acta Math. Sinica 24 (2008), 955-970 (pdf).
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It is a coincidence that I know anything about this. I have been working with Alexander Berkovich on and off for a year or so,

and

Note that there is an entire publication called The Ramanujan Journal on this sort of thing, Alex's department is involved, anyway

http://www.math.ufl.edu/~fgarvan/rama.html

and

http://www.springer.com/mathematics/numbers/journal/11139

with board

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Well, I am more curious on the neighbour positivity Q but have no time now and am already exhausted by the yesterday negativity Q (where the author is not friendly). –  Wadim Zudilin Jun 23 '10 at 2:12