We will present new partition identities that are, in a certain sense, dual to Gordon’s identities. These results arise from a correspondence between three classes of objects: a new family of partitions (called neighborly partitions), monomial ideals, and certain infinite (hyper)graphs. This talk is based on joint works, one with Zahraa Mohsen and another with Pooneh Afsharijoo.
Gaurav Bhatnagar, Atul Dixit, and Krishnan Rajkumar,
The talk next week is by Jehanne Dousse of the University of Geneva. The title and abstract are below.
There was a snafu in the previous talk organisation. Some announcements/reminders were inadvertently not sent. The video of the talk by Ritwik Pal (IIIT, Delhi) has been uploaded on sfandnt website.
Talk Announcement:
Title:Andrews-Gordon-Bressoud type identities and particle motion
Speaker:Jehanne Dousse (University of Geneva)
When: October 16, 2025, 4:00 PM- 5:00 PM IST (12:30 PM CEST)
The Andrews-Gordon identities are among the most important q-series and partition identities, and generalise the famous Rogers-Ramanujan identities. Interestingly, while the product side of these identities clearly corresponds to partitions with congruence conditions, it is not obvious that the sum side of the q-series version is the generating function for the partitions with frequency conditions that appear in the combinatorial version. It was originally proved by George Andrews using recurrences, and then bijectively by Ole Warnaar using particle motion.
In this talk, we will explain and generalise the particle motion approach. We will show that the generalised version can be applied to the sum side of Bressoud's identity and that, using the Andrews-Gordon and Bressoud identities as starting points, it can prove many known and new identities.
This is based on joint work with Jihyeug Jang, Frédéric Jouhet and Isaac Konan.
Gaurav Bhatnagar, Atul Dixit, and Krishnan Rajkumar,
We will present a brief overview of the shifted convolution sum problems, especially those related to the coefficients of automorphic L-functions. Then we will present our recent article on establishing a non-trivial upper bound for the Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke–Maass forms. As a consequence, it gives a significant improvement over the previously known range of shifts for which a non-trivial upper bound of shifted convolution sum holds. This is a joint work with Sampurna Pal.
Gaurav Bhatnagar, Atul Dixit, and Krishnan Rajkumar,
We derive asymptotic expansions for weighted partition numbers satisfying certain conditions. As applications we partially settle some conjectures by Berkovic and Garvan, and by Seo and Yee, on the nonnegativity of the coefficients of certain infinite products, and a conjecture by Chan and Yesilyurt on the periodicity of the signs of the coefficients of a non-theta product. This is joint work with Nian Hong Zhou.
Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)
Abstract:
The minimal excludant of a partition $\pi=(\pi_1,\pi_2,\ldots,\pi_k)$ of $n$ is the smallest positive integer that is not present in $\pi$ and is denoted by $\textup{mex}(\pi)$. The least $r$-gap of $\pi$ is the least positive integer that does not appear in the partition at least $r$ times. In the first half of the talk, we derive some arithmetic functions related to the sum of least $r$-gaps. Using a Tauberian theorem due to Ingham, we obtain Hardy-Ramanujan-type asymptotic formula for two such functions. We also briefly mention some arithmetic properties for these functions.
In the second half, we prove some new identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then
The contents of the talk is taken from two chapters of the speaker's Ph.D. thesis completed under the supervision of Prof. Nayandeep Deka Baruah at Tezpur University, India.
We begin by applying the Laplace transform to derive closed forms for several challenging integrals that seem nearly impossible to evaluate. Using the solution to the Pythagorean equation a^2 + b^2 = c^2, we find that these closed forms become even more intriguing. This approach enables us to propose new integral representations for the error function, with some of the resulting formulas appearing as Fourier sine and cosine transforms.
This week's talk is by Krishnan Rajkumar of Jawaharlal Nehru University (JNU), Delhi. Here is the announcement.
Talk Announcement:
Title:Telescoping continued fractions for several of Ramanujan's entries
Speaker: Krishnan Rajkumar (JNU, Delhi) When: April 10, 2025, 4:00 PM- 5:00 PM IST (Our usual time)
Where: Zoom:
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
Our talk this week is by Jeremy Lovejoy of CNRS. Here is the announcement.
Talk Announcement:
Title:Bailey pairs, radial limits of q-hypergeometric false theta functions, and a conjecture of Hikami
Speaker: Jeremy Lovejoy (CNRS, Paris)
When: Nov 21, 2024, 4:00 PM- 5:00 PM IST (11:30 AM CET)
Where: Zoom: Please write to the organisers
Live LInk: https://youtube.com/live/PCvF5pykt9c?feature=share
Abstract:In the first part of this talk we describe a conjecture of Hikami on the values of the radial limits of a family of q-hypergeometric false theta functions. Hikami conjectured that the radial limits are obtained by evaluating a truncated version of the series. He proved a special case of his conjecture by computing the Kashaev invariant of certain torus links in two different ways. We show how to prove the full conjecture using Bailey pairs. In the second part ofthe talk we explain how the framework of Bailey pairs leads to further results of this type. The talkis based on joint work with Rishabh Sarma.
Best wishes,
Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)
Abstract:The Voronoi summation formulas for the divisor function and $r_2(n)$ are well-known. Not only do these formulas have interesting structure, but they have also been used to improve the error term in the Dirichlet divisor problem and the Gauss circle problem respectively.
In this work we derive Voronoi summation formulas for some other functions related to the generalized divisor function-$d^2(n)$ and Liouville Lambda function-$\lambda(n)$ and Mobius function-$\mu(n)$. We also make use of Vinogradov-Korobov zero free region for the Riemann zeta function to obtain the results.
We also derive beautiful analogues of Cohen's identity and the Ramanujan-Guinand formula associated to these functions.
We also derive certain Omega-bounds for the weighted sums of $d^2(n)$, $\lambda(n)$ and $\mu(n)$ assuming the Linear Independence conjecture.
This is a joint work with Atul Dixit.
Best wishes,
Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)
Using physics methods, Saha and Sinha recently obtained many intriguing expansions of string amplitudes (see Sinha's talk in this seminar 3 October). From a mathematical perspective, they are very unusual deformations of classical hypergeometric identities. One very special case gives a deformation of Madhava's famous series for pi, which received a lot of media attention. I will discuss these identities from a mathematical perspective. They can be derived from partial fraction expansions for symmetric rational functions, which may have some independent interest.
We are back to our usual time with a talk by Aninda Sinha of IISc, Bangalore.
Next week (on Oct 10), we will have a follow-up talk by Hjalmar Rosengren on the topic: String amplitudes and partial fractions: A mathematician's perspective. The formal announcement will be made on the weekend. But please mark your calendars.
Talk Announcement:
Title:Field theory expansions of string theory amplitudes
Speaker:Aninda Sinha (Indian Institute of Science, Bangalore) When: Oct 3, 2024, 4:00 PM- 5:00 PM IST
This talk is based on: Phys. Rev. Lett.132 (2024) 22,221601 (e-print: 2401.05733 [hep-th])
I will explain the reasons (both physics and maths) for trying to find a new formula, satisfying certain physics inspired criteria, for the Euler-Beta and related functions. I will give a sketch of the derivation which relies on a novel dispersion relation. Towards the end, time permitting, new results from the Bootstrap will be presented.
Best wishes,
Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)
