The next speaker in our series is Siddhi Pathak, S. Chowla Research Assistant Professor, Penn State University. The talk announcement is below.

Talk Announcement:

Title:Special values of L-functions

Speaker: Siddhi Pathak (Penn State)

When: May 13, 2021 - 3:55 PM - 5:00 PM (IST)

Where:Zoom (the link will be sent by email to our list)

Live link: https://youtu.be/SXl9IPgE2aI

Tea or Coffee: Please bring your own.

Abstract:In 1730s, Euler resolved the famous Basel problem by evaluating values of the Riemann zeta-function at even positive integers as rational multiples of powers of pi. Thus, we recognize that the values \zeta(2k) are transcendental and algebraically dependent. The situation is drastically different for odd zeta-values, that are not only expected to be transcendental, but also algebraically independent. Although we are far from proving this, there has been striking progress in the work of Apery, and more recently by Ball-Rivoal, Zudilin and others. In this talk, we discuss the analogous problem for Dirichlet L-functions, more generally, Dirichlet series with periodic coefficients.

This talk will be accessible to graduate students.

Where: Zoom: Please write to sf-and-nt@gmail.com for the link.

Tea or Coffee: Please bring your own.

Abstract:We show that the series expansions of certain $q$-products have \textit{matching coefficients} with their inverses. Several of the results are associated to Ramanujan's continued fractions. For example, let $R(q)$ denote the Rogers-Ramanujan continued fraction having the well-known $q$-product repesentation $R(q)=\left(q,q^4;q^5\right)_{\infty}/\left(q^2,q^3;q^5\right)_{\infty}$. If \begin{align*} \sum_{n=0}^{\infty}\alpha(n)q^n=\dfrac{1}{R^5\left(q\right)}=\left(\sum_{n=0}^{\infty}\alpha^{\prime}(n)q^n\right)^{-1},\\ \sum_{n=0}^{\infty}\beta(n)q^n=\dfrac{R(q)}{R\left(q^{16}\right)}=\left(\sum_{n=0}^{\infty}\beta^{\prime}(n)q^n\right)^{-1}, \end{align*} then \begin{align*} \alpha(5n+r)&=-\alpha^{\prime}(5n+r-2), \quad r\in\{3,4\}; \\ \text{and} & \\ \beta(10n+r)&=-\beta^{\prime}(10n+r-6), \quad r\in\{7,9\}. \end{align*} This is a joint work with Hirakjyoti Das.

The next speaker in our seminar is Shishuo Fu of Chongqing University, PRC. It may be Fool's day, but we're not kidding. It really is Shishuo who has consented to give a talk all the way from China!

The live broadcast did not work as anticipated in the previous talk; I hope it works this time. At any rate, its best to try and come for the zoom session.

Talk Announcement

Title: Bijective recurrences for Schroeder triangles and Comtet statistics

Speaker: Shishuo Fu (Chongqing University, PRC)

When: April 1, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom: Please write to sfandnt@gmail.com for a link

Tea or Coffee: Please bring your own.

Abstract:

In this talk, we bijectively establish recurrence relations for two triangular arrays, relying on their interpretations in terms of Schroeder paths (resp. little Schroeder paths) with given length and number of hills. The row sums of these two triangles produce the large (resp. little) Schroeder numbers. On the other hand, it is well-known that the large Schroeder numbers also enumerate separable permutations. This propelled us to reveal the connection with a lesser-known permutation statistic, called initial ascending run (iar), whose distribution on separable permutations is shown to be given by the first triangle as well. A by-product of this result is that "iar" is equidistributed over separable permutations with "comp", the number of components of a permutation. We call such statistics Comtet and we briefly mention further work concerning Comtet statistics on various classes of pattern avoiding permutations. The talk is based on joint work with Zhicong Lin and Yaling Wang.

The next talk is by Christian Krattenthaler. I hope this time the live broadcast works. Here is the announcement.

Talk announcement

Title: Determinant identities for moments of orthogonal polynomials

Speaker: Christian Krattenthaler (University of Vienna, Austria)

When: March 18, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Zoom: Please write to sfandnt@gmail.com for a link

Tea or Coffee: Please bring your own.

Abstract: We present a formula that expresses the Hankel determinants of a linear combination of length d+1 of moments of orthogonal polynomials in terms of a d x d determinant of the orthogonal polynomials. As a literature search revealed, this formula exists somehow hidden in the folklore of the theory of orthogonal polynomials as it is related to "Christoffel's theorem". In any case, it deserves to be better known and be presented correctly and with full proof. (During the talk I will explain the meaning of these somewhat cryptic formulations.) Subsequently, I will show an application of the formula. I will close the talk by presenting a generalisation that is inspired by Uvarov's formula for the orthogonal polynomials of rationally related densities.

We are happy to report that Atul Dixit, one of the co-organizers of this seminar, has been awarded the 2021 GÃ¡bor SzegÃ¶ Prize. This prize is awarded every two years by the SIAM Activity Group on Orthogonal Polynomials and Special Functions
(SIAG/OPSF). It is awarded to an
early-career researcher for outstanding research contributions within 10 years of obtaining a Ph.D.

The
selection committee for the 2021 award consists of Peter Clarkson
(Chair), University of Kent; Kerstin Jordaan, University of South
Africa; Adri Olde Daalhuis, The University of Edinburgh; Sarah Post,
University of Hawaii; and Yuan Xu, University of Oregon.

The selection committee in its letter to him cited his “impressive scientific work solving problems related to number theory using special functions, in particular related to the work of Ramanujan.”

Atul obtained his Ph.D. under the direction of Bruce Berndt in 2012 from the University of Illinois at Urbana-Champaign. Subsequently he did a post-doc at Tulane with Victor Moll as his mentor. Currently, he is in IIT, Gandhinagar and has quickly developed a reputation among young and upcoming mathematicians in this country that has attracted a bright set of Ph.D. students and post-docs to his team.

We wish Atul continued success, both personally and for the group he is leading.

Gaurav Bhatnagar and Krishnan Rajkumar (co-organizers with Atul of this seminar).

Abstract: We study the parity of coefficients of classical mock theta functions. Suppose $g$ is a formal power series with integer coefficients, and let $c(g;n)$ be the coefficient of $q^n$ in its series expansion. We say that $g$ is of parity type $(a,1-a)$ if $c(g;n)$ takes even values with probability $a$ for $n\geq 0$. We show that among the 44 classical mock theta functions, 21 of them are of parity type $(1,0)$. We further conjecture that 19 mock theta functions are of parity type $(\frac{1}{2},\frac{1}{2})$ and 4 functions are of parity type $(\frac{3}{4},\frac{1}{4})$. We also give characterizations of $n$ such that $c(g;n)$ is odd for the mock theta functions of parity type $(1,0)$.

Where: Google Meet: Please write to sfandnt@gmail.com for a link.

Tea or Coffee: Please bring your own.

ABSTRACT

We
develop a calculus that gives an elementary approach to enumerate
partition-like objects using an infinite upper-triangular
number-theoretic matrix. We call this matrix the Partition-Frequency
Enumeration (PFE) matrix. This matrix unifies a large number of results
connecting number-theoretic functions to partition-type functions. The
calculus is extended to arbitrary generating functions, and functions
with Weierstrass products. As a by-product, we recover (and extend) some
well-known recurrence relations for many number-theoretic functions,
including the sum of divisors function, Ramanujan's $\tau$ function,
sums of squares and triangular numbers, and for $\zeta(2n)$, where $n$
is a positive integer. These include classical results due to Euler,
Ramanujan, and others. As one application, we embed Ramanujan's famous
congruences $p(5n+4)\equiv 0\;$ (mod $5)$ and $\tau(5n+5)\equiv 0\; $
(mod $5)$ into an infinite family of such congruences.

The next talk will be by Victor Moll of Tulane University. This will be the first mathematician from the US giving a talk in our seminar. Victor has kindly consented to stay awake to make his talk more suitable for Indian timings. But in future, we do expect speakers from the US will speak at times later at night (IST).

At any rate, we hope more speakers from the US will give talks in our seminar. As we have mentioned earlier, our website now contains video recordings of the presentations. This makes it more convenient for the US participants to view the talks.

Talk Announcement

Title: Valuations of interesting sequences

Speaker: Victor Moll (Tulane)

When: February 4, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Google Meet; Please write to sfandnt@gmail.com if you want a link.

Tea or Coffee: Please bring your own.

ABSTRACT

Given a sequence ${ a_{n} }$ of integers and a prime $p$, the sequence of

valuation $\nu_{p}(a_{n})$ presents interesting challenges. This talk will discuss a

variety of examples in order to illustrate these challenges and present our approach

Our next talk is on January 21, 2021. Please do feel free to volunteer to speak, if you have submitted something recently and the topic is suitable for this group.

We have now upgraded the sf-and-nt website a bit. In case a video recording is available, we have embedded it, so you can view the recording at leisure. I wish to thank Shivam Sahu, a recent graduate of the math department of Ashoka University, for help in this activity. In particular, it will be of help when you are reading the associated papers in more detail. Any suggestions for the website are welcome.

The talk for the week is as follows.

Talk Announcement

Title: Elliptic and q-analogs of the Fibonomial numbers

Speaker: Josef KÃ¼stner (University of Vienna)

When: January 21, 2021 - 3:55 PM - 5:00 PM (IST)

Where: Google Meet; Please write to sfandnt@gmail.com if you want a link.

Tea or Coffee: Please bring your own.

ABSTRACT

The Fibonomial numbers are integer numbers obtained from the binomial coefficients by replacing each term by its corresponding Fibonacci number. In 2009, Sagan and Savage introduced a simple combinatorial model for the Fibonomial numbers.

In this talk, I will present a combinatorial description for a q-analog and an elliptic analog of the Fibonomial numbers which is achieved by introducing certain q- and elliptic weights to the model of Sagan and Savage.

This is joint work with Nantel Bergeron and Cesar Ceballos.

A very happy new year to all. We have decided that the first talk
of every year will be a Ramanujan Special Talk. This year a colloquium
talk will be given by Wadim Zudilin. The announcement is below.

We
wish you many new theorems, ideas and papers in 2021. Please do send
any ideas or suggestions you have for the organisers to make this
seminar more successful and help serve the interests of this community.

Talk Announcement

Title: 10 years of q-rious positivity. More needed!

Date and Time: Thursday, January 7, 2021, 3:55 PM IST (GMT+5:30)

Tea or coffee: Bring your own.

Where: Zoom: Please write to sfandnt@gmail.com for a link at-least 24 hours before the talk.

Abstract:

The $q$-binomial coefficients \[ \prod_{i=1}^m(1-q^{n-m+i})/(1-q^i),\] for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the many combinatorial interpretations of them. Ten years ago, together with Ole Warnaar we observed that this non-negativity (aka positivity) property generalises to products of ratios of $q$-factorials that happen to be polynomials; we prove this observation for (very few) cases. During the last decade a resumed interest in study of generalised integer-valued factorial ratios, in connection with problems in analytic number theory and combinatorics, has brought to life new positive structures for their $q$-analogues. In my talk I will report on this "$q$-rious positivity" phenomenon, an ongoing project with Warnaar.

For the last seminar of the year, we are proud to host Rahul Kumar. He has recently defended his thesis in IIT, Gandhinagar under the direction of Atul Dixit, one of the co-organizers of this seminar. The announcement appears below.

Talk Announcement

Title: A generalized modified Bessel function and explicit transformations of certain Lambert series

Speaker: Rahul Kumar (IIT, Gandhinagar)

When: December 17, 2020 - 3:55 PM - 5:00 PM (IST)

Where: Google Meet; Please write to sfandnt@gmail.com if you want a link.

Tea or Coffee: Please bring your own.

ABSTRACT:

An exact transformation, which we call a master identity, is obtained for the series $\sum_{n=1}^{\infty}\sigma_{a}(n)e^{-ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. As corollaries when $a$ is an odd integer, we derive the well-known transformations of the Eisenstein series on $\text{SL}_{2}\left(\mathbb{Z}\right)$, that of the Dedekind eta function as well as Ramanujan's famous formula for $\zeta(2m+1)$. Corresponding new transformations when $a$ is a non-zero even integer are also obtained as special cases of the master identity. These include a novel companion to Ramanujan's formula for $\zeta(2m+1)$. Although not modular, it is surprising that such explicit transformations exist. The Wigert-Bellman identity arising from the $a=0$ case of the master identity is derived too. The latter identity itself is derived using Guinand's version of the Vorono\"{\dotlessi} summation formula and an integral evaluation of N.~S.~Koshliakov involving a generalization of the modified Bessel function $K_{\nu}(z)$. Koshliakov's integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new transformation involving the sums-of-squares function $r_k(n)$. This is joint work with Atul Dixit and Aashita Kesarwani.

The following are the conferences happening all over the country in December (mostly in honor of Ramanujan's birthday). You may forward this email to everyone in our group.

1.RMS & Rajagiri school of engineering and Technology:(Dec. 11-14, 2020)

The next talk in the SF and NT seminar is by Sneha Chaubey. There will be one further talk in the Seminar this year, next week, and then we will take a short break.

Here is the announcement for Sneha's talk.

Talk announcement

Title: Generalized visible subsets of two dimensional integer lattice

Speaker: Sneha Chaubey (IIIT, Delhi)

When: December 10, 2020 - 3:55 PM - 5:00 PM (IST)

Where: Google Meet; Please write to sfandnt@gmail.com if you want a link.

Tea or Coffee: Please bring your own.

ABSTRACT:

We will discuss some subsets of two-dimensional integer lattice which arise as visible sets under some suitable notion of visibility. We will discuss some set-theoretic (Delone, Meyer, Quasicrystals etc.), geometrical (density and gaps) and dynamical (auto-correlation and diffraction pattern) properties of these subsets.

The next talk is by Manjil Saikia of Cardiff University. The remaining two talks of the year will be on December 10 and December 17.

We will begin the new year with a special public lecture by Wadim Zudilin. More details as we come nearer 2021.

Talk announcement

Title: : Refinedenumerationofsymmetry classes of Alternating Sign Matrices

Speaker: Manjil P. Saikia (Cardiff University)

When: November 26, 2020 - 3:55 PM - 5:00 PM (IST)

Where: Google Meet; Please write to sfandnt@gmail.com if you want a link.

Tea or Coffee: Please bring your own.

ABSTRACT:

The sequence $1,1,2,7,42,429, \ldots$ counts several combinatorial objects, some of which I will describe in this talk. The major focus would be one of these objects, alternating sign matrices (ASMs). ASMs are square matrices with entries in the set {0,1,-1}, where non-zero entries alternate in sign along rows and columns, with all row and column sums being 1. I will discuss some questions that are central to the theme of ASMs, mainly dealing with their enumeration. In particular we shall prove some conjectures of Fischer, Robbins, Duchon and Stroganov. This talk is based on joint work with Ilse Fischer and some ongoing work.