Live LInk: https://youtube.com/live/eVgu2PvtG7U?feature=share
Abstract
We will recall earlier work where Apéry's proof of irrationality of ζ(3)was related to a continued fraction in Ramanujan's notebooks. We will then recall the method of Telescoping continued fractions from joint work with Bhatnagar (2023). We will then proceed to apply this method to certain series to prove several entries from Ramanujan's notebooks related toπ, ζ(2), ζ(3)andG,the Catalan's constant.
Best wishes,
Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)
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, wewill 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.
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.
The talk next week will be by James Sellers of the University of Minnesota, Duluth. We are back to our usual time, since the speaker is currently in Europe.
Talk Announcement:
Title:Partitions into Odd Parts with Designated Summands
Speaker: James Sellers (University of Minnesota, Duluth, USA) When: Feb 20, 2025, 4:30 PM- 5:30 PM IST (12 Noon CET) (Note the time).
In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also studied such partitions wherein all parts must be odd, and they denoted the number of such partitions of size n by the function PDO(n).
In this talk, I will report on recent proofs of infinite families of divisibility properties satisfied by PDO(n). Some of these proofs follow in elementary fashion while others rely on modular forms (in work completed jointly with Shane Chern).
I will then transition to very recent joint work with Shishuo Fu in which we consider a "refined" view of PDO(n) based on ideas which originated with P. A. MacMahon. This new approach allows for a more combinatorial view of the well-known identity that, for all n,
$$PDO(2n) = \sum_{0 \leq k \leq n} PDO(k) PDO(n-k),$$
a result which is (trivially) proven via generating functions but which has eluded combinatorial proof for many years.
The first talk of the year (on Feb 6, 2025) is a ``Ramanujan Special". This year's speaker is Krishnaswami Alladi. Among other things, he is the founding editor of the Ramanujan Journal. Please note that the talk will be later than usual.
Talk Announcement: The 2025 Ramanujan Special
Title:Speaker: Krishnaswami Alladi (University of Florida, USA)
When: Feb 6, 2025, 6:30 PM- 7:30 PM IST (8 AM EST) (Note special time)
(EST= IST - 10:30)
Where: Zoom: Write to sfandnt@gmail.com for a link
In 1977, I noticed two duality identities connecting the smallest and largest prime factors of integers, and vice-versa, the connection being provided by the Moebius function. Using this duality, I generalized the famous results of Edmund Landau on the Moebius function which are equivalent to the Prime Number Theorem. When this duality is combined with the Prime Number Theorem for Arithmetic Progressions, this leads to the striking result that
