Archna Kumari (IIT, Delhi) -- Thursday 27 Mar, 2025 -- 4:00 PM

 Dear all,

This week we have a talk by Archna Kumari of IIT, Delhi. Here is the announcement. 

Talk Announcement: 

Title: Some results in weighted and elliptic enumeration

Speaker: Archna Kumari (IIT, Delhi)
When: Mar 27, 2025, 4:00 PM- 5:00 PM IST (Our usual time)

Where: Zoom: Write to sf and nt at gmail dot com for the link

Live LInk: https://youtube.com/live/6U6er9-X7ZE?feature=share

Abstract

In the literature, there are a lot of $q$-identities. In this talk, we will talk about two types of $q$-identities and their extension to the elliptic case. First, we extend some Fibonacci identities using combinatorial methods. Since many of these identities can be derived through telescoping, we use this technique to find elliptic versions of basic elementary identities, such as the sum of the first odd or even numbers, the geometric series sum, and the sum of the first cubes. Along the way, we discover a multi-parameter identity that seems to be new, even in the q-setting. Second, we will extend some $q$-hypergeometric identities to elliptic hypergeometric. We derive four expansion formulas and, as a result, some transformation formulas. In the $ q$ case, when the nome $p=0$, one of the formulas generalizes the basic hypergeometric transformation formula due to Liu, and Wang and Ma. The remaining three are equivalent to the well-poised Bailey lemma. Thus, we recover transformation formulas from Warnaar and Spiridonov.

This is joint work with Gaurav Bhatnagar and Michael Schlosser.

Atul Dixit (IIT, Gandhinagar) - Thursday, Mar 6, 2025 - 4:00 PM

 Dear all,

Next week's talk is by Atul Dixit of IIT, Gandhinagar. Here is the announcement. 

Talk Announcement: 

Title: The Rogers-Ramanujan dissection of a theta function

Speaker: Atul Dixit (IIT, Gandhinagar)
When: Mar 6, 2025, 4:00 PM- 5:00 PM IST (Our usual time)

Where: Zoom: Write to the organisers for the link

Live Link: https://youtube.com/live/1v54wzJk-_o?feature=share

Abstract

Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which, as shown by Andrews, not only gives a famous modular relation between the Rogers-Ramanujan functions $G(q)$ and $H(q)$ as a corollary but also a relation between two fifth order mock theta functions and $G(q)$ and $H(q)$. In this talk, we will discuss a generalization of Ramanujan's relation that we recently obtained which gives an infinite family of such identities. Our result shows that a theta function can always be ``dissected'' as a finite sum of products of generalized Rogers-Ramanujan functions. 

Several well-known results are shown to be consequences of our theorem, for example, a generalization of the Jacobi triple product identity and Andrews' relation between two of his generalized third order mock theta functions. As will be shown, the identities resulting from our main theorem for $s>2$ transcend the modular world and hence look difficult to be written in the form of a modular relation. Using asymptotic analysis, we also offer compelling evidence that explains how Ramanujan may have arrived at his generalized modular relation. This is joint work with Gaurav Kumar.