What is the relationship between modular forms and the Rogers-Ramanujan identities? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T18:20:33Z http://mathoverflow.net/feeds/question/29117 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29117/what-is-the-relationship-between-modular-forms-and-the-rogers-ramanujan-identitie What is the relationship between modular forms and the Rogers-Ramanujan identities? Vladimir Sotirov 2010-06-22T16:51:07Z 2010-06-23T09:01:57Z <p>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. </p> <p>Let H(q) be the generating function for partitions with the same condition plus that 1 is not a part.</p> <p>These are the left-hand sides of the Rogers-Ramanujan Identities.</p> <blockquote> <p><code>$G(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2}}{(q;q)_n}=\frac{1}{(q;q^5)_\infty(q^4;q^5)_\infty}$</code></p> <p><code>$H(q)=\displaystyle\sum_{n=0}^\infty\frac{q^{n^2+n}}{(q;q)_n}=\frac{1}{(q^2;q^5)_\infty(q^3;q^5)_\infty}$</code></p> </blockquote> <p>I am intrigued by the following unreferenced statement in the wikipedia <a href="http://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_identities#Modular_functions%20%22page%22" rel="nofollow">page</a>:</p> <blockquote> <p>If q = e<sup>2πiτ</sup>, then q<sup>−1/60</sup>G(q) and q<sup>11/60</sup>H(q) are modular functions of τ.</p> </blockquote> <ol> <li><p>Do modular forms shed any light on the Rogers-Ramanujan Identities, or is the connection (as far as we know) a curious coincidence?</p></li> <li><p>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.</p></li> <li><p>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)?</p></li> </ol> http://mathoverflow.net/questions/29117/what-is-the-relationship-between-modular-forms-and-the-rogers-ramanujan-identitie/29130#29130 Answer by Will Jagy for What is the relationship between modular forms and the Rogers-Ramanujan identities? Will Jagy 2010-06-22T18:13:53Z 2010-06-22T18:13:53Z <p>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,</p> <p><a href="http://www.math.ufl.edu/~alexb/" rel="nofollow">http://www.math.ufl.edu/~alexb/</a> </p> <p>and </p> <p><a href="http://www.math.ufl.edu/fac/facmr/Berkovich.html" rel="nofollow">http://www.math.ufl.edu/fac/facmr/Berkovich.html</a> </p> <p>Note that there is an entire publication called The Ramanujan Journal on this sort of thing, Alex's department is involved, anyway</p> <p><a href="http://www.math.ufl.edu/~fgarvan/rama.html" rel="nofollow">http://www.math.ufl.edu/~fgarvan/rama.html</a></p> <p>and</p> <p><a href="http://www.springer.com/mathematics/numbers/journal/11139" rel="nofollow">http://www.springer.com/mathematics/numbers/journal/11139</a></p> <p>with board</p> <p><a href="http://www.springer.com/mathematics/numbers/journal/11139?detailsPage=editorialBoard" rel="nofollow">http://www.springer.com/mathematics/numbers/journal/11139?detailsPage=editorialBoard</a> </p> http://mathoverflow.net/questions/29117/what-is-the-relationship-between-modular-forms-and-the-rogers-ramanujan-identitie/29176#29176 Answer by SandeepJ for What is the relationship between modular forms and the Rogers-Ramanujan identities? SandeepJ 2010-06-23T00:11:15Z 2010-06-23T00:11:15Z <p>At the root, these identities arise because there exist theta function identities (e.g. <a href="http://en.wikipedia.org/wiki/Jacobi_triple_product" rel="nofollow">Jacobi triple product</a>) 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 <a href="http://en.wikipedia.org/wiki/Bailey_pair" rel="nofollow">Bailey lemma</a> also comes up in this context.</p> <p>David Bressoud in his book <em>Analytic and combinatorial generalizations of the Rogers-Ramanujan identities</em> 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.</p> <p>Generalizations have been proved - see <a href="http://mathworld.wolfram.com/Goellnitz-GordonIdentities.html%20" rel="nofollow">Gordon-Gollnitz identities</a> and <a href="http://mathworld.wolfram.com/Andrews-GordonIdentity.html" rel="nofollow">Andrews-Gordon identity</a></p> <p>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.</p> <p><a href="http://www.math.uiuc.edu/~berndt/publications.html" rel="nofollow">http://www.math.uiuc.edu/~berndt/publications.html</a></p> <ol> <li>Partition identities and Ramanujan's modular equations (with N. D. Baruah), J. Comb. Thy. (A) 114 (2007), 1024-1045 (pdf).</li> <li>Partition identities arising from theta function identities (with N. D. Baruah), Acta Math. Sinica 24 (2008), 955-970 (pdf).</li> </ol> http://mathoverflow.net/questions/29117/what-is-the-relationship-between-modular-forms-and-the-rogers-ramanujan-identitie/29185#29185 Answer by Wadim Zudilin for What is the relationship between modular forms and the Rogers-Ramanujan identities? Wadim Zudilin 2010-06-23T01:30:36Z 2010-06-23T01:30:36Z <p>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 <a href="http://arxiv.org/abs/1001.1571" rel="nofollow">contribution</a>, 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.</p> <p>As for the original question,</p> <blockquote> <p>What is the relationship between Modular Forms and the Rogers-Ramanujan Identities?</p> </blockquote> <p>the answer is straightforward: whenever you see Rogers-Ramanujan-type identities, both sides are <em>modular</em> 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.</p>