Ramanujan Explained - Lecture 1 - April 18, 2024, 4:00 PM- 5:00 PM IST

 Dear all,

We have launched a course under the title of Ramanujan Explained. There will be a series of lectures, all given by Gaurav Bhatnagar, with accompanying notes and exercises. The goal is to cover (a large number of) Ramanujan's identities. The first talk in this series is in our next seminar slot. Kindly do share this announcement with students who may be interested in Ramanujan and his mathematics. The first few lectures will target $q$-hypergeometric series and special cases, and can serve as an introduction to basic hypergeometric series. We hope these lectures will serve as a useful supplement to the monumental work of Bruce Berndt  (Ramanujan's Notebooks I-V) and George Andrews and Bruce Berndt  (Ramanujan's Lost Notebook I-V).  

Lecture notes, slides, lecture videos and more: Ramanujan Explained Website

We will not be including the notifications of these talks on this website. 

Talk Announcement: 

Title:  Ramanujan Explained 1: How to discover the Rogers-Ramanujan identities

Speaker:  Gaurav Bhatnagar (Ashoka University)
When: April 18, 2024, 4:00 PM- 5:00 PM IST 

Where: Zoom: Write to the organisers (sfandnt at gmail dot com) for the link

Live LInk: https://youtube.com/live/_M4WbbAffIA?feature=share

Abstract:

About the Rogers-Ramanujan identities, Hardy famously remarked: "It would be difficult to find more beautiful formulae than the "Rogers-Ramanujan" identities... " In the first introductory lecture in the Ramanujan Explained course, we explain Askey's idea on how Ramanujan may have come across these identities. Continued fractions played an important part of Ramanujan's work, and Askey's explanation is all about the simplest $q$-continued fraction and how it naturally leads to the Rogers-Ramanujan identities.

Gaurav Bhatnagar (Ashoka University) -- Thursday, Mar 21, 2024 - 4:00 PM (IST)

 Dear all,

The next talk is by Gaurav Bhatnagar of Ashoka University. The announcement is as follows. 

Talk Announcement: 

Title:  Elliptic enumeration and identities

Speaker:  Gaurav Bhatnagar (Ashoka University)
When: Mar 21, 2024, 4:00 PM- 5:00 PM IST 

Where: Zoom: Please write to the organisers for the link.

Live LInk: https://youtube.com/live/cpqHK-R2oXg?feature=share

Abstract

Many of the ideas of $q$-counting and $q$-hypergeometrics are now being extended to the elliptic case. The approach is not very far from the $q$-case. In this talk, we show several examples to illustrate this idea. First we extend some Fibonacci identities using combinatorial methods. Many such identities can be found by telescoping, so we next use telescoping to find elliptic extensions of elementary identities such as the sum of the first  odd or even numbers, the geometric sum and the sum of the first  cubes. In the course of our study, we obtained an identity with many parameters, which appears to be new even in the $q$-case. Finally, we introduce elliptic hypergeometric series and give an extension of some important identities of Liu. As applications, we find 5 double summations and 4 new elliptic transformation formulas. Again, these are new in the $q$-hypergeometric case, where the nome $p$ is 0. 

This is a report of joint work with Archna Kumari and Michael Schlosser. 

Gaurav Bhatnagar - Talk 2 on Thursday, April 21, 2022

On the coming Thursday, Atul Dixit (IIT, Gandhinagar) and Gaurav Bhatnagar (Ashoka University) will present the talks they presented at the recently concluded JMM meeting held online. Each talk will be approximately 20 minutes followed by 10 minutes for questions. The titles and abstracts are below.

Here is the announcement for the second talk of the day.

Title: An easy proof of  Ramanujan's famous mod $5$ congruences

Speaker: Gaurav Bhatnagar (Ashoka University)

When: Thursday, April 21, 2022 - 4:30 PM - 5:00 PM

Where: Zoom. 

Live LInk: https://youtu.be/IB3FiBM2j5s

Abstract 

We give an elementary proof of Ramanujan's famous congruences $p(5n+4) \equiv 0 \mod 5$ and $\tau(5n+5)\equiv 0 \mod 5$. The proof requires no more than Euler's techniques and a result of Jacobi. The proof extends to embed the congruences into 4 infinite families of congruences for rational powers of the eta function. 

This is joint work with Hartosh Singh Bal.

The video below is the same as the one in Atul's talk. It just begins in the middle. 

Gaurav Bhatnagar (Ashoka) February 18, 2021 - 3:55 PM-5:00 PM

 

Talk announcement

Title: The Partition-Frequency Enumeration Matrix

Speaker: Gaurav Bhatnagar (Ashoka University)

When: February 18, 2021 - 3:55 PM - 5:00 PM (IST)

 

Where: Google Meet:  Please write to sfandnt@gmail.com for a link.

Tea or Coffee: Please bring your own. 

ABSTRACT

We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Partition-Frequency Enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-type functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan's $\tau$ function, sums of squares and triangular numbers, and for $\zeta(2n)$, where $n$ is a positive integer. These include classical results due to Euler, Ramanujan, and others.  As one application, we embed Ramanujan's famous congruences $p(5n+4)\equiv 0\;$ (mod $5)$ and $\tau(5n+5)\equiv 0\; $ (mod $5)$ into an infinite family of such congruences.

This is joint work with Hartosh Singh Bal. 

Gaurav Bhatnagar, SPS, JNU

Title: Ramanujan's $q$-continued fractions

Speaker: Gaurav Bhatnagar 

When: Tuesday, January 14, 2020; 4:00--5:00 pm,   

Where: Seminar Room, Indian Statistical Center, 7 SJS Sansanwal Marg, Delhi 110016

ABSTRACT

 It is exactly a hundred years since Ramanujan died at an early age of 32 years, but his mathematical legacy lives on. One of the things he was famous for was his work on continued fractions. We will give a brief introduction to Ramanujan's life and the material available on his story and his work. We will show Ramanujan's $q$-continued fractions and show how an elementary idea used by Euler can be used to prove many of Ramanujan's continued fractions. We expect to briefly mention some topics of current research interest. Students are welcome; much of the talk will be suitable for a general audience.  

***

A page from Ramanujan's Lost Notebook with some continued fractions

PDF version of this letter

Gaurav Bhatnagar, SPS, JNU

Title: How to discover the Rogers-Ramanujan identities
Speaker: Gaurav Bhatnagar, SPS, JNU


Where: Seminar Room, School of Physical Sciences (SPS)

Abstract:
The Rogers-Ramanujan identities were sent by Ramanujan to Hardy in a letter more than a 100 years ago. In the next few years, the identities were circulated among mathematicians, but nobody, including Ramanujan, could prove them. Then one day, while riffling through old back copies of the journal, Ramanujan discovered them in an obscure paper written in 1894 by the English mathematician Rogers. Later, these identities were discovered independently by Schur in a combinatorial context, and then again in 1980 by Baxter in the context of mathematical physics.  We don’t know how Ramanujan got to them, but we examine a method to conjecture these identities which Askey has suggested may be the way Ramanujan discovered them.