Friday, June 24, 2022

Neelam Kandhil (IMSc, Chennai) - Thursday, June 30, 2022 - 4:00 PM - 5:00 PM (IST)

We are delighted to welcome Neelam Kandhil  (soon to become Dr. Neelam Kandhil) of IMSc, Chennai for our next talk.  The talk announcement is below.

Talk Announcement:

Title:  On an extension of a question of Baker

SpeakerNeelam Kandhil (IMSc, Chennai)
 
When: Thursday, June 30, 2022 - 4:00 PM - 5:00 PM (IST) 

Where: Zoom. Please ask the organisers for a link

Live Link: https://youtu.be/A89gu25J_zs

Tea or Coffee: Please bring your own.

Abstract:  

It is an open question of Baker whether the numbers $L(1, \chi)$ for non-trivial Dirichlet characters $\chi$ with period $q$ are linearly independent over $\mathbb{Q}$. The best known result is due to Baker, Birch and Wirsing which affirms this when $q$ is co-prime to $\varphi(q)$.  In this talk, we will discuss an extension of this result to any arbitrary family of moduli. The interplay between the resulting ambient number fields brings in new technical issues and complications hitherto absent in the context of a fixed modulus. In the process, we also extend a result of Okada about linear independence of the cotangent values over $\mathbb{Q}$ as well as a result of Murty-Murty about ${\overline{\mathbb{Q}}}$ linear independence of such $L(1, \chi)$ values.

If time permits, we will also discuss the interrelation between the non-vanishing of Dedekind zeta values and its derivatives.

Monday, May 30, 2022

Arindam Roy (UNC, Charlotte) - Thursday, June 2, 2022 - 3:55 PM

 Our next speaker is Arindam Roy of the University of North Carolina at Charlotte. The talk announcement is below.

Talk Announcement:

Title: On the Hyperbolicity of Jensen Polynomials for Power Partitions.

Speaker: Arindam Roy (UNC, Charlotte)
 
When: Thursday, June 2, 2022 - 4:00 PM - 5:00 PM (IST) 

Where: Zoom

Tea or Coffee: Please bring your own.

Abstract:  

Polya established that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function. This led to the study of Jensen polynomials for large classes of functions. Recently, Jensen polynomials for various partition functions and more generally, the Jensen polynomials for weakly holomorphic forms are considered. Their hyperbolicity was established under various conditions. In this talk we will consider the Jensen polynomials for power partitions and discuss the hyperbolicity of these Jensen polynomials

Wednesday, May 18, 2022

Amita Malik (Max Plank Institute, Bonn) - Thursday, May 19, 2022 - 4:00 PM - 5:00 PM (IST)

 Our next speaker is Amita Malik of the Max Plank Institute. The talk announcement is below.

Talk Announcement:

Title: Partitions into primes in arithmetic progressions

Speaker: Amita Malik (Max Plank Institute, Bonn)
 
When: Thursday, May 19, 2022 - 4:00 PM - 5:00 PM (IST) 

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

Tea or Coffee: Please bring your own.

Abstract:  

 In this talk, we discuss the asymptotic behavior of the number of partitions into primes in arithmetic progressions. While the study for ordinary partitions goes back to Hardy and Ramanujan, partitions into primes were recently re-visited by Vaughan. Our error term is sharp and improves on previous known estimates in the special case of primes as parts of the partition. As an application, monotonicity of this partition function is established explicitly via an asymptotic formula.

Saturday, April 30, 2022

Murali K Srinivasan (IIT Bombay) - Thursday, May 5, 2022

Our next speaker is Murali Srinivasan of IIT Bombay (at Mumbai). The talk announcement is below.

Talk Announcement:

Title: A q-analog of the adjacency matrix of the n-cube.

Speaker: Murali K Srinivasan (IIT Bombay)
 
When: Thursday, May 5, 2022 - 4:00 PM - 5:00 PM (IST) 

Where: Zoom

Tea or Coffee: Please bring your own.

Abstract:  

Random walk on the $n$-cube gives rise to two matrices, the adjacency matrix and the Kac matrix, that have an elegant spectral theory with many applications.

We define a random walk on subspaces that yields q-analogs of the adjacency and Kac matrices with an equally elegant spectral theory (though more difficult to prove). We give an application to tree counting.

This is joint work with Subhajit Ghosh.

Monday, April 18, 2022

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




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. 

Atul Dixit (IIT Gandhinagar) - Talk 1 on Thursday April 21, 4:00 PM - 4:30 PM

 Dear all,


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.

We should say that during the summer, the talks in the seminar may be less as we are finally able to catch up with colleagues in person. 

Here is the announcement for Atul's talk.

Title:  Combinatorial identities associated with a bivariate generating function for overpartition pairs

Speaker: Atul Dixit (IIT, Gandhinagar)

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


Where: Zoom



Abstract

We obtain a three-parameter $q$-series identity that generalizes two results of Chan and Mao. By specializing our identity, we derive new results of combinatorial significance in connection with $N(r, s, m, n)$, a function counting certain overpartition pairs recently introduced by Bringmann, Lovejoy and Osburn. For example, one of our identities gives a closed-form evaluation of a double series in terms of Chebyshev polynomials of the second kind, thereby resulting in an analogue of Euler's pentagonal number theorem. Another of our results expresses a multi-sum involving $N(r, s, m, n)$ in terms of just the partition function $p(n)$. Using a result of Shimura we also relate a certain double series with a weight 7/2 theta series. This is joint work with Ankush Goswami.

Saturday, March 19, 2022

Soumyarup Banerjee (IIT, Gandhinagar) - Thursday, Mar 24, 2022 - 3:55 PM

Our next speaker is Soumyarup Banerjee of IIT, Gandhinagar. The talk announcement is below.

Talk Announcement:

Title: Finiteness theorems with almost prime inputs

Speaker: Soumyarup Banerjee (IIT, Gandhinagar)
 
When: Thursday, Mar 24, 2022 - 4:00 PM - 5:00 PM (IST) 

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

Live Link: https://youtu.be/Fd2gNRyK04I

Tea or Coffee: Please bring your own.

Abstract:  

The Conway–Schneeberger Fifteen theorem states that a given positive definite integral quadratic form is universal (i.e., represents every positive integer with integer inputs) if and only if it represents the integers up to 15. This theorem is sometimes known as “Finiteness theorem" as it reduces an infinite check to a finite one. In this talk, I would like to present my recent work along with Ben Kane where I have investigated quadratic forms which are universal when restricted to almost prime inputs and have established finiteness theorems akin to the Conway–Schneeberger Fifteen theorem.


Saturday, March 5, 2022

Wenguang Zhai (China Inst. of Mining and Tech, Beijing, PRC) - Thursday, Mar 10, 3:55 PM (IST)

Our next speaker of Wenguang Zhai of the China Institute of Mining and Technology, Beijing, who will speak on recent work with Xiadong Cao (Beijing Institute of Petro-Chemical Technology, Beijing) and Yoshio Tanigawa (Nagoya, Japan).

Talk Announcement:


Title: The MC-algorithm and continued fraction formulas involving ratios of Gamma functions

Speaker: Wenguang Zhai (China Institute of Mining and Technology, Beijing, PRC)
 
When: Thursday, Mar 10, 2022 - 4:00 PM - 5:00 PM (IST) 

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

Live Link: https://youtu.be/C6jzvD2NPuo

Tea or Coffee: Please bring your own.

Abstract:  

Ramanujan discovered many continued fraction expansions about ratios of the Gamma functions. However, Ramanujan left us no clues about how he discovered these elegant formulas. In this talk, we will explain the so-called MC-algorithm. By this algorithm, we can not only rediscover many of Ramanujan's continued fraction expansions, but also find some new formulas.

Sunday, February 20, 2022

Krishnan Rajkumar (JNU) - Thursday Feb 24, 2022 - 3:55 PM (IST)

The next talk is by Krishnan Rajkumar. This talk will be recorded but not available online immediately.  


Talk Announcement:

Title: The Binet function and telescoping continued fractions

Speaker: Krishnan Rajkumar (Jawaharlal Nehru University (JNU))
 
When: Thursday, Feb 24, 2022 - 4:00 PM - 5:00 PM (IST) 

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

Tea or Coffee: Please bring your own.

Abstract:  

The Binet function $J(z)$ defined by the equation $\Gamma(z)  = \sqrt{2 \pi} z^{z-\frac{1}{2}}e^{-z} e^{J(z)}$ is a well-studied function. The Stirling approximation comes from the property $J(z) \rightarrow 0$ as $z\rightarrow \infty$, $|arg z|<\pi$. In fact, an asymptotic expansion $J(z) \sim z^{-1} \sum_{k=0}^{\infty} c_k z^{-2k}$ holds in this region, with closed form expressions for $c_k$ and explicit integrals for the error term for any finite truncation of this asymptotic series. 

In this talk, we will discuss two different classical directions of research. The first is exemplified by the work of Robbins (1955) and Cesaro (1922), and carried forward by several authors, the latest being Popov (2018), where elementary means are used to find rational lower and upper bounds for $J(n)$ which hold for all positive integers $n$. All of these establish inequalities of the form $J(n)-J(n+1) > F(n)-F(n+1)$ for an appropriate rational function $F$ to derive the corresponding lower bounds by telescoping.

The second direction is to use moment theory to derive continued fractions of specified forms for $J(x)$. For instance, a modified S-fraction of the form $\frac{a_1}{x \ +} \frac{a_2}{x \ +}\frac{a_3}{x \ +} \cdots$ can be formally derived from the above asymptotic expansion using a method called the qd-algorithm. The resulting continued fraction can then be shown to converge to $J(x)$ by the asymptotic properties of $c_k$ and powerful results from moment theory. There are no known closed-form expressions for the $a_k$.

We will then outline what we call the method of telescoping continued fractions to extend the elementary methods of the first approach to derive the modified S-fraction for $J(x)$ obtained in the second by a new algorithm. We will describe several results that we can prove and some conjectures that together enhance our understanding of the numbers $a_k$ as well as provide upper and lower bounds for $J(x)$ that improve all known results.

This is joint work with Gaurav Bhatnagar.
 ---

Gaurav Bhatnagar (Ashoka), Atul Dixit (IIT, Gandhinagar) and Krishnan Rajkumar (JNU)

Saturday, February 5, 2022

Surbhi Rai (IIT, Delhi) - Thursday, Feb 10, 2022 - 3:55 PM IST

 Dear all,


The next talk is by Surbhi Rai, a graduate student in IIT, Delhi. The announcement is as follows. 

Talk Announcement:

Title: Expansion Formulas for Multiple Basic  Hypergeometric Series Over Root Systems

Speaker: Surbhi Rai (IIT, Delhi)
 
When: Thursday, Feb 10, 2022 - 4:00 PM - 5:00 PM (IST) 

Where: Zoom. Please write to the organizers for the link.
 
Live Link: https://youtu.be/3WlOJivytek

Tea or Coffee: Please bring your own.

Abstract:  In a series of works, Zhi-Guo Liu extended some of the central summation and transformation formulas of basic hypergeometric series. In particular, Liu extended Rogers' non-terminating very-well-poised  $_{6}\phi_{5}$  summation formula, Watson's transformation
formula, and gave an alternate approach to the orthogonality of the Askey-Wilson polynomials. These results are helpful in number-theoretic contexts too. All this work relies on three expansion formulas of Liu. 

This talk will present several infinite families of extensions of Liu's fundamental formulas to multiple basic hypergeometric series over root systems. We will also discuss results that extend Wang and Ma's generalizations of Liu's work which they obtained using $q$-Lagrange inversion. Subsequently, we will look at an application based on the expansions of infinite products. These extensions have been obtained using the $A_n$ and $C_n$ Bailey transformation and other summation theorems due to Gustafson, Milne, Milne and Lilly, and others, from $A_n$, $C_n$ and $D_n$ basic hypergeometric series theory. We will observe how this approach brings Liu's expansion formulas within the Bailey transform methodology. 

This talk is based on joint work with Gaurav Bhatnagar. (https://arxiv.org/abs/2109.02827)

 ---

Gaurav Bhatnagar (Ashoka), Atul Dixit (IIT, Gandhinagar) and Krishnan Rajkumar (JNU)

www.sfnt.org

sfandnt@gmail.com

Saturday, January 22, 2022

Ramanujan Special: Alan Sokal (University College, London and New York University) Thursday, January 27, 2022, 4-5 PM (IST)

Dear all, 

Welcome to 2022. We begin the year with a Ramanujan Special talk by Alan Sokal. The talk announcement is below. 

We encourage you to distribute this announcement to friends and colleagues in your department or otherwise, so that they come to know of our seminar. 

Talk Announcement:

Title: Coefficientwise Hankel-total positivity

Speaker: Alan Sokal (University College London and New York)
 
When: Thursday, January 27,  2022 - 4:00 PM - 5:00 PM (IST) 

Where: Zoom: Write to sfandnt@gmail.com for the link


Tea or Coffee: Please bring your own.

Abstract:  

  A matrix $M$ of real numbers is called totally positive
   if every minor of $M$ is nonnegative.  Gantmakher and Krein showed
   in 1937 that a Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$
   of real numbers is totally positive if and only if the underlying
   sequence $(a_n)_{n \ge 0}$ is a Stieltjes moment sequence.
   Moreover, this holds if and only if the ordinary generating function
   $\sum_{n=0}^\infty a_n t^n$ can be expanded as a Stieltjes-type
   continued fraction with nonnegative coefficients:
$$
   \sum_{n=0}^{\infty} a_n t^n
   \;=\;
   \cfrac{\alpha_0}{1 - \cfrac{\alpha_1 t}{1 - \cfrac{\alpha_2 t}{1 -  \cfrac{\alpha_3 t}{1- \cdots}}}}
$$
   (in the sense of formal power series) with all $\alpha_i \ge 0$.
   So totally positive Hankel matrices are closely connected with
   the Stieltjes moment problem and with continued fractions.

   Here I will introduce a generalization:  a matrix $M$ of polynomials
   (in some set of indeterminates) will be called
   coefficientwise totally positive if every minor of $M$
   is a polynomial with nonnegative coefficients.   And a sequence
   $(a_n)_{n \ge 0}$ of polynomials will be called
   coefficientwise Hankel-totally positive if the Hankel matrix
   $H = (a_{i+j})_{i,j \ge 0}$  associated to $(a_n)$ is coefficientwise
   totally positive.  It turns out that many sequences of polynomials
   arising naturally in enumerative combinatorics are (empirically)
   coefficientwise Hankel-totally positive.  In some cases this can
   be proven using continued fractions, by either combinatorial or
   algebraic methods;  I will sketch how this is done.  In many other
   cases it remains an open problem.

   One of the more recent advances in this research is perhaps of
   independent interest to special-functions workers:
   we have found branched continued fractions for ratios of contiguous
   hypergeometric series ${}_r \! F_s$ for arbitrary $r$ and $s$,
   which generalize Gauss' continued fraction for ratios of contiguous
   ${}_2 \! F_1$.  For the cases $s=0$ we can use these to prove
   coefficientwise Hankel-total positivity.

   Reference: Mathias P\'etr\'eolle, Alan D.~Sokal and Bao-Xuan Zhu,
   arXiv:1807.03271