Monday, December 4, 2023

David Bradley (Maine) - Thursday Dec 7, 2023 - 6:30 PM (IST) (NOTE. Special Time)

 Dear all,

The talk this week is by David Bradley of the University of Maine. Please note the special time. Since Professor Bradley is located in the US, we are starting later than usual. 

This will be the final talk of the year. We will come back next year with a Ramanujan special, and hope that we get an opportunity to meet in person in the upcoming conference and holiday season. We wish you happy holidays and a great new year. 

The announcement is as follows. 

Talk Announcement: 

Title:    On Fractal Subsets of Pascal's "Pyramid" and the Number of  Multinomial Coefficients Congruent to a Given Residue Modulo a Prime

Speaker: David Bradley (University of Maine, USA)

When: Thursday, Thursday Dec 7 23, 2023 - 6:30 PM (IST) (8AM EST/ 2PM (CET))

Where: Zoom: Ask the organisers for a link

We obtain an explicit formula and an asymptotic formula for the number of multinomial coefficients which are congruent to a given residue modulo a prime, and which arise in the expansion of a multinomial raised to any power less than a given power of that prime. Each such multinomial coefficient can be associated with a certain Cartesian product of intervals contained in the unit cube. For a fixed prime, the union of these products forms a set which depends on both the residue and the power of the prime. In the limit as the power of the prime increases to infinity, the sequence of unions converges in the Hausdorff metric to a non-empty compact set which is independent of the residue. We calculate the fractal dimension of this limiting set, and consider its monotonicity properties as a function of the prime.  To study the relative frequency of various residue classes in the sequence of approximating sets, it would be desirable to have a closed-form formula for the number of entries in the first p rows of Pascal’s "pyramid" which are congruent to a given nonzero residue r modulo the prime p. Unfortunately, numerical computations with large prime moduli suggests that if there is such a formula, it is extremely complicated. Nevertheless, the evidence indicates that for sufficiently large primes p, the number of binomial coefficients in Pascal's triangle which are congruent to r mod p for r = 1, 1 < r < p1, and r p1 is well approximated by the respective linear functions of p given 
by 3pp/2, and pIn particular, for large primes p there are approximately six times as many occurrences of the residue 1 in the first p rows of Pascal’s triangle reduced modulo p than there are of any other residue in the range 1 < r < p − 1, and three times as many as r p − 1. On the other hand, if we let the nonnegative integer k vary while keeping the prime p fixed, and look at the relative frequency of various residue classes that occur in the first pk rows, the seemingly substantial differences in frequency between r = 1, 1 < r < p1, and p1 when k = 1 are increasingly dissipated as k grows without bound. We show that in the limit as k tends to infinity, all nonzero residues are equally represented with asymptotic proportion 1/(p − 1).

Sunday, November 19, 2023

Sonika Dhillon (ISI, Delhi) - Thursday, Thursday Nov 23, 2023 - 4:00 PM (IST)

 Dear all,

The talk this week is by Sonika Dhillon, ISI, Delhi. The announcement is as follows. 

Talk Announcement: 
Title:   Linear independence of numbers
Speaker: Sonika Dhillon (ISI, Delhi)
When: Thursday, Thursday Nov 23, 2023 - 4:00 PM (IST) 

Where: Zoom: Write to the organisers for the link

Let $\psi(x)$ denote the digamma function that is the logarithmic derivative of $\Gamma$ function.
In 2007, Murty and Saradha studied the linear independence of special values of  digamma function $\psi(a/q)+\gamma$ over some specific numbers fields which also imply the  non-vanishing of $L(1,f)$ for any rational-valued Dirichlet type function $f$.  In 2009, Gun, Murty and Rath studied the non-vanishing of $L'(0,f)$ for even Dirichlet-type periodic $f$ in terms of $L(1,\hat{f})$ and established that this is related to the linear independence of logarithm of gamma values.  In this direction, they made a conjecture which they call it as a variant of Rohrlich conjecture concerning the linear independence of logarithm of gamma values. In this talk, first we will discuss the  linear independence of digamma values over the field of algebraic numbers. Later, we provide counterexamples
to this variant of  Rohrlich conjecture.

Sunday, November 5, 2023

Sagar Shrivastava (TIFR, India) - Thursday, Thursday Nov 9, 2023 - 4:00 PM (IST)

 Dear all,

The talk in the coming week is by Sagar Shrivastava, School of Mathematics, Tata Institute of Fundamental Research (TIFR).

Talk Announcement: 
Title:   Representations, Determinants and Branching rules
Speaker: Sagar Shrivastava (TIFR, India)
When: Thursday, Thursday Nov 9, 2023 - 4:00 PM (IST) 

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

Branching rules/laws (restriction of representations) also known as symmetry breaking in physics has been an active area of research since the onset of the topic by Herman Weyl in 1950. In this talk, I would give a brief description of Highest weight theory and the determinantal form of the Weyl character formula. I would proceed to talk about branching from $GL_n$ to $GL_{n-1}$ and give an idea about the other classical groups. 

Sunday, October 22, 2023

Seamus Albion (Vienna, Austria) - Thursday Oct 26, 2023 - 4:00 PM (IST)

 Dear all,

The next talk is by Seamus Albion of the University of Vienna. The announcement is as follows.

Talk Announcement: 
Title:   An elliptic $A_n$ Selberg integral
Speaker: Seamus Albion (Vienna, Austria)
When: Thursday, Thursday Oct 26, 2023 - 4:00 PM (IST) (12:30 PM CEST)
Where: Zoom: Write to the organisers to get the link

Selberg's multivariate extension of the beta integral appears 
all over mathematics: in random matrix theory, analytic number theory,  multivariate orthogonal polynomials and conformal field theory. The goal of my talk will be to explain a recent unification of two important generalisations of the Selberg integral, namely the Selberg integral associated with the root system of type A_n due to Warnaar and the elliptic Selberg integral conjectured by van Diejen and Spiridonov and proved by Rains. The key tool in our approach is the ellipticinterpolation kernel, also due to Rains. This is based on joint work with Eric Rains and Ole Warnaar.

Saturday, October 7, 2023

David Wahiche (Univeriste de Tours, France) - Thursday, October 12, 2023 - 4:00 PM (IST)

 Dear all,

The next talk is by David Wahiche of the University of Tours, France. The title and abstract is below. 

Talk Announcement: 
Title:   From Macdonald identities to Nekrasov--Okounkov type formulas
Speaker: David Wahiche (Universite' de Tours, France)
When: Thursday, Oct 12, 2023, 4:00 PM- 5:00 PM IST 
Where: Zoom: Ask the organisers for a link


Between 2006 and 2008, using various methods coming from representation theory (Westbury), gauge theory (Nekrasov--Okounkov) and combinatorics (Han), several authors proved the so-called Nekrasov–Okounkov formula which involves hook lengths of integer partitions.

This formula does not only cover the generating series for P, but more generally gives a connection between powers of the Dedekind η function and integer partitions. Among the generalizations of the Nekrasov--Okounkov formula, a (q, t)-extension was proved by Rains and Warnaar, by using refined skew Cauchy-type identities for Macdonald polynomials. The same result was also obtained independently by Carlsson–Rodriguez-Villegas by means of vertex operators and the plethystic exponential. As mentioned in both of these papers, the special case q=t of their formula correspond to a q version of the Nekrasov--Okounkov formula, which was already obtained by Dehaye and Han (2011) and Iqbal et al. (2012).

Motivated by the work of Han et al. around the generalizations of the Nekrasov--Okounkov formula, one way of deriving Nekrasov--Okounkov formula is by using the Macdonald identities for infinite affine root systems (Macdonald 1972), which can be thought as extension of the classical Weyl denominator formula.

In this talk, I will try to explain how some reformulations of the Macdonald identities (Macdonald 1972, Stanton 1989, Rosengren and Schlosser 2006) can be decomposed in the basis of characters for each infinite of the 7 infinite affine root systems by the Littlewood decomposition. This echoes a representation theoretic interpretation of the Macdonald identities (see the book of Carter for instance) and an ongoing project with Cédric Lecouvey, I will mention some partial results we get.

At last, I will briefly explain how to go from these reformulations of Macdonald identities to q Nekrasov--Okounkov type formulas.

Monday, September 25, 2023

Seema Kushwaha (IIIT, Allahabad) - Thursday Sept 28, 2023 - 4:00 PM (IST)

Dear all, sorry for the late announcement. The next talk is by Seema Kushwaha of IIIT, Allahabad. We are back to our usual time now. Hope to see you later this week. 

Talk Announcement: 
Title:   Farey-subgraphs and Continued Fractions
Speaker: Seema Kushwaha (IIIT, Allahabad)
When: Thursday, Sept 28, 2023, 4:00 PM- 5:00 PM IST 
Where: Zoom: Please send email to the organisers for a link.

Let $p$ be a prime and $l\in\mathbb{N}$. Let \begin{equation*}\label{X_n}
\mathcal{X}_{p^l}=\left\{\frac{x}{y}:~x,y\in\mathbb{Z},~ y>0,~\mathrm{gcd}(x,y)=1~\textnormal{and}~{p^l}|y\right\}\cup\{\infty\}.
The set $\mX_{p^l}$ is the vertex set of a connected graph where vertices $x/y$ and $u/v$ are adjacent if and only if $ xv-uy=\pm p^l.$ These graphs give rise to a family of continued fraction, namely, $\f_{p^l}$-continued fractions \cite{seema_fareysubgraphs}.

  Let $\mathcal{X}$ be a  subset of the extended set of rational numbers. A {\it best $\mathcal{X}$-approximation} of a real number is a  notion which is analogous to best rational approximation. 

An element  $u/v$ of $\mX$ is called a \textit{best $\mX$-approximation} of $x\in\R$, if for every $u'/v'\in\mX$ different from $u/v$ with $0< v' \le v$, we have $|vx-u|<|v'x-u'|$.  
In  this talk, we will discuss the existence and uniqueness of $\f_{p^l}$-continued fractions and their approximation properties. 

Thursday, September 14, 2023

Shashank Kanade (University of Denver) - Thursday Sept 14, 2023 - 6:00 PM (IST)

 Dear all,

We are back after an extended summer break. I hope many of us had an opportunity to meet each other and further our research goals. 

The next talk is by Shashank Kanade, University of Denver. It is a little later in the evening from our usual time.

Talk Announcement: 

On the $A_2$ Andrews--Schilling--Warnaar identities

Speaker: Shashank Kanade (University of Denver)

When: Thursday, Sept 14, 2023, 6:00 PM- 7:00 PM IST (6:30 AM MDT)

Where: Zoom: please write to the organisers for the link

I will give a description of my work with Matthew C. Russell on the $A_2$
Andrews--Schilling--Warnaar identities. Majority of our single variable
sum=product conjectures have been proven by S. O. Warnaar; I will also
explain what remains. Bi-variate versions of our conjectures are largely open. 

Sunday, May 21, 2023

Michael Schlosser (Vienna, Austria) - Thursday May 25, 2023 - 4:00 PM (IST)

 Dear all,

The next talk is by Michael Schlosser of the University of Vienna, Austria. 

After this talk we will be taking a break for the summer. Hopefully, we will get an opportunity to meet in person during this time. 

Talk Announcement: 

Bilateral identities of the Rogers-Ramanujan type

Speaker: Michael Schlosser (University of Vienna, Austria)
When: May 25, 2023, 4:00 PM- 5:00 PM IST (12:30 PM CEST)
Where: Zoom. Please write to the organisers for a link

The first and second Rogers-Ramanujan (RR) identities have a prominent history. They were originally discovered and proved in 1894 by Leonard J. Rogers, and then independently rediscovered by the legendary self-taught Indian mathematician Srinivasa Ramanujan some time before 1913. They were also independently discovered and proved in 1917 by Issai Schur. About the RR identities Hardy remarked

`It would be difficult to find more beautiful formulae than the
``Rogers-Ramanujan'' identities, ...'

Apart from their intrinsic beauty, the RR identities have served as a stimulus for tremendous research around the world. The RR and related identities have found interpretations in
various areas including combinatorics, number theory, probability theory, statistical mechanics, representations of Lie algebras, vertex algebras, knot theory and conformal field theory.

In this talk, a number of bilateral identities of the RR type will be presented. We explain how these identities can be derived by analytic means using identities for bilateral basic hypergeometric series. Our results include bilateral extensions of the RR and
of the Göllnitz-Gordon identities, and of related identities 
by Ramanujan, Jackson, and Slater.

Corresponding results for multiseries are given as well, including multilateral extensions of the Andrews-Gordon identities, of Bressoud's even modulus identities, and others.

This talk is based on the speaker's preprint arXiv:1806.011153v2 (which has been accepted for publication in Trans. Amer. Math. Soc.).

Sunday, May 7, 2023

Bishal Deb (University College, London) - Thursday May 11, 2023 - 4:00 PM (IST)

 Dear all,

The next talk is by Bishal Deb of University College, London. Here is the announcement. 

Talk Announcement: 

The "quadratic family" of continued fractions and combinatorial sequences

Speaker: Bishal Deb (University College, London)
When: May 11, 2023, 4:00 PM- 5:00 PM IST (11:30 AM BST)
Where: Zoom:

We will begin this talk by introducing some combinatorial sequences whose Stieltjes-type continued fraction coefficients increase linearly. We briefly mention the work of Sokal and Zeng where they systematically studied multivariate generalisations of these continued fractions for factorials, Bell numbers and double factorials. 

Next, we will define the Genocchi and median Genocchi numbers and  introduce D-permutations, a class of permutations which enumerate these numbers. We mention some multivariate continued fractions counting various statistics on D-permutations.

Finally, we move to the secant numbers and introduce cycle-alternating permutations; these are another class of permutations  which enumerate the secant numbers. We mention some multivariate continued fractions counting various statistics on cycle-alternating permutations. We then describe the entries in the Jacobi-Rogers matrix of our continued fraction using alternating Laguerre digraphs, which are a class of directed graphs. If time permits, we will briefly state some remarks on the Jacobian elliptic functions.

This talk will be based on joint work with Alan Sokal.

Tuesday, April 25, 2023

Rahul Kumar (Penn State University) - Thursday, April 27 - 4:00 PM

The next talk is by Rahul Kumar, Fulbright-Nehru Postdoctoral Fellow, Penn State University.  The announcement is as follows. 

Talk Announcement: 

Title: Arithmetic properties of the Herglotz-Zagier-Novikov function

Speaker: Rahul Kumar (Penn State University)
When: Apr 27, 2023, 4:00 PM- 5:00 PM IST 
Where: Zoom: Write to for a link
Live Link:


The Kronecker limit formulas are concerned with the constant term in the Laurent series expansion of certain Dirichlet series at $s=1$. Various special functions appear in Kronecker limit formulas; one of them is \emph{Herglotz function}. Recently, Radchenko and Zagier extensively studied the properties of the Herglotz function, such as its special values, connection to Stark's conjecture, etc. This function appeared in the work of Herglotz, and Zagier. After providing an overview of the history of this research area, we will discuss the arithmetic properties of a Herglotz-type function that appears in a Kronecker limit formula derived by Novikov. For example, we will present the two- and three-term functional equations satisfied by it along with its special values. This is joint work with Professor YoungJu Choie.

Friday, April 7, 2023

A. Sankaranarayanan (Hyderabad) - April 13, 2023 - 4:00 PM

 Dear all,

The next talk is by A. Sankaranarayanan of the School of Mathematics and Statistics, University of Hyderabad. The announcement is as follows. 

Talk Announcement: 

Title: On the Rankin-Selberg L-function related to the Godement-Jacquet L-function

Speaker: A. Sankaranarayanan (University of Hyderabad)
When: Apr 13, 2023, 4:00 PM- 5:00 PM IST 
Where: Zoom: Write to organisers for the link
We discuss the Riesz mean of the Coefficients of the Rankin-Selberg L-function related to the Godement-Jacquet L-function.

This is a joint work with Amrinder Kaur and recently appeared in Acta Mathematica Hungarica.

Wednesday, March 29, 2023

Christophe Vignat (Tulane) - Thursday, Mar 30 - 4:00 PM

 The next talk is by Christophe Vignat.   It will be at our usual time of 4 PM IST. 

Talk Announcement: 

Title: Dirichlet Series Under Standard Convolutions: Variations on Ramanujan’s Identity for Odd Zeta Values 

Speaker: Christophe Vignat (Université Paris-Saclay, CentraleSupelec, Orsay, France and Tulane University)
When: Mar 30, 2023, 4:00 PM- 5:00 PM IST (1:30 PM EEST)
Where: Zoom. Write to the organisers for a link.


I will show a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; this formula is inspired by a famous identity of Ramanujan.  Some specializations of this convolution formula produce new identities and allow to recover several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by A. Dixit and his collaborators, or the convolution of Bernoulli numbers by Y. Komori and his collaborators. 

If time permits, I will also exhibit some matrix product representations for the Riemann zeta function evaluated at even and odd integers.

This is joint work with P. Chavan, S. Chavan and T. Wakhare.

Saturday, March 11, 2023

Bruce Berndt (UIUC) - Thurs Mar 16 - 6:00 PM (Note Special Time)

 Dear all,

We are happy to announce our next talk is by Professor Bruce Berndt, the world's biggest authority on Ramanujan's mathematics and related areas. 

Please note the special time. It is two hours later than usual. Please circulate this announcement in your department. 

Please see a further announcement below.

Talk Announcement: 

Title: Finite Trigonometric Sums: Evaluations, Estimates, Reciprocity Theorems

Speaker: Bruce Berndt (University of Illinois at Urbana Champaign)
When: Mar 16, 2022, 6:00 PM- 7:00 PM IST (7:30 AM - 8:30 AM (CDT))
Where: Zoom. Please write to the organisers for the link.

First, motivated by a theorem in Ramanujan's lost notebook, Martino Fassina, Sun Kim, Alexandru Zaharescu, and the speaker developed representations for finite sums of products of trig functions for which we provided theorems and several conjectures.   

Second, a paper of Richard McIntosh served as motivation.  First, he made a very interesting conjecture, which was recently proved by Likun Xie, Zaharescu, and the speaker.  Second, he examined a particular trigonometric sum, which inspired Sun Kim, Zaharescu, and the speaker to evaluate in closed form several classes of trigonometric sums, and find reciprocity theorems for others.  

New conference announcement
A new conference on algebraic combinatorics has been announced by Arvind Ayyer. It is called 
Meru Annual Combinatorics Conference
Dates: 29th to 31st May, 2023

Sunday, February 26, 2023

B. Ramakrishnan, ISI, Tezpur - Thursday, Mar 2, 2023 - 4:00 PM

The next talk is by B. Ramakrishnan (popularly known as Ramki), formerly of HRI, Allahabad, and now in ISI, Tezpur. 

Talk Announcement: 

Title: An extension of Ramanujan-Serre derivative map and some applications.

Speaker: B. Ramakrishnan (Indian Statistical Institute North-East Center, Tezpur)
When: Mar 2, 2022, 4:00 PM- 5:00 PM IST 
Where: Zoom: Please write to organisers for the link.


In this talk, we present a simple extension of the Ramanujan-Serre derivative map and 
describe how it can be used to derive a general method for explicit evaluation of convolution sums  of the divisor functions. We provide explicit examples for four types of convolution sums.

This is a joint work with Brundaban Sahu and Anup Kumar Singh.  

Saturday, February 11, 2023

Galina Filipuk, University of Warsaw - Thursday, Feb 16, 2023 - 4:00 PM

We are back to our usual time with a talk by Galina Filipuk all the way from Warsaw, Poland. Please note that we will be open to changing the time, since speakers from the US find this time to be very inconvenient, and we surely would like speakers from the US. The discussions in the previous talk went quite late into the night (for New Zealand) and we thank Shaun Cooper for a very nice talk. 

Talk Announcement: 

Title: (Quasi)-Painleve equations and Painleve equivalence problem

Speaker: Galina Filipuk (University of Warsaw, Poland)
When: Feb 16, 2023, 4:00 PM- 5:00 PM IST (11:30 CET in Warsaw)
Where: Zoom: Please write to for the link.


Painleve equations are second order nonlinear differential equations solutions of which have no movable critical points (algebraic singularities). They appear in many applications (e.g., in the theory of orthogonal polynomials) but in disguise. How to find a transformation to the canonical form? This is known as the Painleve equivalence problem.
The so-called geometric approach may help in many cases.

In this talk I shall present some recent results on the geometric approach for the Painleve and quasi-Painleve equations.

Saturday, January 28, 2023

Ramanujan Special: Shaun Cooper (Massey University) - Thursday, Feb 2, 2023 - 2:30 PM

 Happy new year. 

The first talk of the year (on Feb 2, 2023) is a "Ramanujan Special". This year's speaker is Shaun Cooper. Please note that the talk will be earlier than usual.  

The last year was quite exciting for our group with many talks as well as a mini course. We hope this year is equally exciting. Please consider the seminar to present your latest preprint. 

Talk Announcement: Ramanujan Special

Title: Apéry-like sequences defined by four-term recurrence relations: theorems and conjectures

Speaker: Shaun Cooper (Massey University, Auckland, New Zealand)

When: Feb 2, 2022, 2:30 PM- 3:30 PM IST (Note special time) (IST= GMT - 5:30)

Where: Zoom. Write to for a link.


The Apéry numbers are famous for having been introduced and used by R. Apéry to prove that~$\zeta(3)$ is irrational. They may be defined by the recurrence relation
with the single initial condition $A(0)=1$ being enough to start the recurrence. The Apéry numbers are all integers, a fact not obvious from the recurrence relation, and they satisfy interesting congruence properties. The generating function
y=\sum_{n=0}^\infty A(n)w^n
has a splendid parameterisation given by
y = \prod_{j=1}^\infty \frac{(1-q^{2j})^7(1-q^{3j})^7}{(1-q^{j})^5(1-q^{6j})^5}
w=q\,\prod_{j=1}^\infty \frac{(1-q^{j})^{12}(1-q^{6j})^{12}}{(1-q^{2j})^{12}(1-q^{3j})^{12}}.
In this talk I will briefly survey other sequences defined by three-term recurrence relations that have properties similar to those satisfied by the Apéry numbers described above. I will also introduce some sequences defined by four-term recurrence relations and describe some of their properties.

Several conjectures will be presented.

Here are the slides of the talk.