Sunday, August 11, 2019

On Entry II.16.12 of Ramanujan

In June, 2019, I attended a very inspiring conference, held at the University of Illinois at Urbana-Champaign. This was to celebrate Bruce Berndt's 80th birthday and his retirement. Bruce is the greatest living expert on Ramanujan and one of the nicest people in the world. He is scholarly and an inspiration. His retirement means nothing, he is already traveling around the world and he said he is working on another book with Ae Ja Yee on partitions. My wishes to him were more for myself than for him: Wishing us many, many books and papers from you for many, many years!

Mourad Ismail and I have written a paper for the occasion. I presented it in Illinois in front of a small audience, which included Bruce. This is the second paper with Mourad on the subject of Ramanujan's continued fractions, using techniques he has taught me.

Here is a picture from the conference. (More pictures below)

Picture in front of Altgeld Hall, the iconic department of mathematics, UIUC


Here is the announcement of the paper from ArXiv. Click on the link in the title to get to the preprint on ArXiv.
Title: On Entry II.16.12: A continued fraction of Ramanujan
Authors: Gaurav Bhatnagar and Mourad E. H. Ismail
Categories: math.CA
Comments: 15 Pages
MSC-class: 33D45 (Primary), 30B70 (Secondary)
Abstract:
We study a continued fraction due to Ramanujan, that he recorded as Entry 12
in Chapter 16 of his second notebook. It is presented in Part III of Berndt's
volumes on Ramanujan's notebooks. We give two alternate approaches to proving Ramanujan's Entry 12, one using a method of Euler, and another using the theory of orthogonal polynomials. We consider a natural generalization of Entry 12 suggested by the theory of orthogonal polynom
ials.
Here is a picture of Bruce Berndt's return gift to all of us.

Bruce's return gift. The daily presence of Ramanujan in my life

I presented this work in the OPSFA2019 conference in Hagenberg, Austria. Here are the slides from the talk in Hagenberg. OPSFA2019 was an amazing conference. I got a chance to hear Christian's concert on the church organ, saw and heard Chihara, met Alan Sokal (among others),  hung out with Michael and Hjalmar and worked with Mourad on this paper and on our next joint paper. Here are some more pictures from Hagenberg.

The group photo from OPSFA