Schur polynomials are the characters of irreducible representations of classical groups of type A parametrized by partitions. For a fixed integer $t \geq 2$ and a primitive $t$'th root of unity \omega, Schur polynomials evaluated at elements $\omega^{k} x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$, were considered by D. J. Littlewood (AMS press, 1950)and independently by D. Prasad (Israel J. Math., 2016). They characterized partitions for which the specialized Schur polynomials are nonzero and showed that if the Schur polynomial is nonzero, it factorizes into characters of smaller classical groups of type A.

In this talk, I will present a generalization of the factorization result to the characters of classical groups of type B, C and D. We give a uniform approach for all cases. The proof uses Cauchy-type determinant formulas for these characters and involves a careful study of the beta sets of partitions. This is joint work with A. Ayyer and is available here. (Preprint: https://arxiv.org/abs/2109.11310)

The talk this week is by Nicholas Smoot from the Research Institute of Symbolic Computation (RISC) at Johannes Kepler University (JKU), Linz, Austria. The announcement is below.

Since Ramanujan's groundbreaking work, a large variety of infinite congruence families for partition functions modulo prime powers have been discovered. These families vary enormously with respect to the difficulty of proving them. We will discuss the application of the localization method to proving congruence families by walking through the proof of one recently discovered congruence family for the counting function for 5-elongated plane partitions. In particular, we will discuss a critical aspect of such proofs, in which the associated generating functions of a given congruence family are members of the kernel of a certain linear mapping to a vector space over a finite field. We believe that this approach holds the key to properly classifying congruence families.

The non-vanishing as well as primitivity of the values of the Fourier coefficients of non-CM Hecke eigen forms, in particular the Ramanujan $\tau$ function is a deep and mysterious theme in Number theory. In this talk, we will report on our recent work on the number of distinct prime factors of the values of the Fourier coefficients of non-CM Hecke eigen forms, in particular the Ramanujan $\tau$ function.

We are happy that Kaneenika Sinha (IISER, Pune) has consented to give a mini-course on Central limit theorems in number theory. The course will comprise two lectures. The announcement is below. Graduate students who are interested in number theory are especially welcome to hear Professor Sinha.

Mini-course announcement

Title: Central Limit theorems in Number Theory

Speaker: Kaneenika Sinha (IISER, Pune)

Abstract:

The goal of these lectures is to review a theme that binds the study of different types of arithmetic functions, namely central limit theorems. After reviewing the "prototype" theorem in this theme, namely the classical Erdos-Kac theorem about the prime-omega function, we will survey different types of central limit theorems in the context of zeroes of zeta functions, eigenvalues of Hecke operators acting on spaces of cusp forms and eigenvalues of regular graphs.

After a refreshing break in a summer with many opportunities to meet in person, we are back with a talk by Professor Ole Warnaar of the University of Queensland, Australia.

Talk Announcement:

Title: Cylindric partitions and Rogers--Ramanujan identities

Speaker: S. Ole Warnaar (University of Queensland, Australia)

Cylindric partitions, first introduced by Gessel and Krattenthaler in 1997, are an affine (or toroidal) analogue of ordinary plane partitions. Perhaps somewhat surprisingly, cylindric partitions have a close connection to Rogers--Ramanujan-type $q$-series identities. In this talk I will try to explain this connection, and report on some new Rogers--Ramanujan identities for the affine Lie algebra $\mathrm{A}_2^{(1)}$ that follow naturally from this connection.

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.

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

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.

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.

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

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.

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

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.

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.

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)

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.

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.

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

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)

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

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