In the previous lecture in the series, we considered infinite products and the $q$-Gamma function. Now we consider an important product called the theta products, that were considered by Jacobi and are involved in his famous triple product identity. We obtain it as a special case of a result of Ramanujan.
I began including some of the background material, and I will continue with bits and pieces of that as we go along. This may mean somewhat haphazard development of the background material. But we want the focus to be on Ramanujan’s identities. So the background material is motivated by what is required to understand Ramanujan’s entries.
Talk Announcement:
Title: Ramanujan Explained 6: the $_1_\psi_1$ summation
Speaker: Gaurav Bhatnagar (Ashoka University)
When: Sept 26, 2024, 4:00 PM- 5:30 PM IST
Abstract:We give a proof of Ramanujan’s famous bilateral sum for a $_1\psi_1$ sum. The proof is Ismail’s celebrated proof. While it is the proof from `The Book’, it is not Ramanujan’s own proof. It contains Jacobi’s triple product identity and has many corollaries giving product forms of Ramanujan’s theta functions.
Ramanujan Explained 