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

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

2010-06-22T16:51:07Z

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^−1/60G(q) and q^11/60H(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)?

Answer by Will Jagy for What is the relationship between modular forms and the Rogers-Ramanujan identities?

2010-06-22T18:13:53Z

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,

http://www.math.ufl.edu/~alexb/

and

http://www.math.ufl.edu/fac/facmr/Berkovich.html

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

http://www.springer.com/mathematics/numbers/journal/11139?detailsPage=editorialBoard

Answer by SandeepJ for What is the relationship between modular forms and the Rogers-Ramanujan identities?

2010-06-23T00:11:15Z

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

Partition identities and Ramanujan's modular equations (with N. D. Baruah), J. Comb. Thy. (A) 114 (2007), 1024-1045 (pdf).
Partition identities arising from theta function identities (with N. D. Baruah), Acta Math. Sinica 24 (2008), 955-970 (pdf).

Answer by Wadim Zudilin for What is the relationship between modular forms and the Rogers-Ramanujan identities?

2010-06-23T01:30:36Z

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.