Two identities in Ramanujan’s Lost Notebook & Bressoud’s conjecture
In this talk, we discuss two topics. The first topic is about two identities that Ramanujan recorded without proofs in his lost notebook. These two identities are intimately connected with the classical Circle and Divisor problems in number theory, respectively. They involve doubly infinite series of Bessel functions, and in each case, there are three possible interpretations for the double series. We proved the first identity under all three interpretations, and the second under two of them. Furthermore, we discuss many analogues and generalizations of them. This is joint work with Bruce C. Berndt and Alexandru Zaharescu.
The second topic is about Bressoud’s conjecture. In 1980, Bressoud obtained an analytic generalization of the Rogers-Ramanujan-Gordon identities. He then tried to establish a combinatorial interpretation of his identity, which specializes to many well-known Rogers-Ramanujan type identities. He proved that a certain partition identity follows from his identity in a very restrictive case and conjectured that the partition identity holds true in general. We discuss Bressoud’s conjecture for the general case and bijective proofs of it.