## Saturday, August 28, 2021

### Howard Cohl (NIST) Thursday, Sept 2, 2021, 6:30 PM IST

Dear all,

The next talk is by Howard Cohl of NIST. Please note the special time. Howard is zooming in from California, and we are grateful to him to be able to speak at a time suitable to us.

Talk Announcement:

Title: The utility of integral representations for the Askey-Wilson polynomials and their symmetric sub-families

Speaker: Howard Cohl (NIST)

When: Thursday, September 2, 2021 - 6:30 PM - 7:30 PM (IST) (6 am Pacific Day Time (PDT))

Abstract:

The Askey-Wilson polynomials are a class of orthogonal polynomials which are symmetric in four free parameters which lie at the very top of the q-Askey scheme of basic hypergeometric orthogonal polynomials. These polynomials, and the polynomials in their subfamilies, are usually defined in terms of their finite series representations which are given in terms of terminating basic hypergeometric series. However, they also have nonterminating, q-integral, and integral representations. In this talk, we will explore some of what is known about the symmetry of these representations and how they have been used to compute their important properties such as  generating functions. This study led to an extension of interesting contour integral representations of sums of nonterminating basic hypergeometric functions initially studied by Bailey, Slater, Askey, Roy, Gasper and Rahman. We will also discuss how these contour integrals are deeply connected to the properties of the symmetric basic hypergeometric orthogonal polynomials.

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

sfandnt@gmail.com

## Thursday, August 19, 2021

### Rajat Gupta (IIT, Gandhinagar) - Thursday, Aug 19, 2021 - 3:55 PM - 5:00 PM (IST)

The next talk is by Rajat Gupta -- or shall we say Dr. Rajat Gupta!

Here is the announcement.

Talk Announcement:

Title: Koshliakov zeta functions and modular relations
Speaker: Rajat Gupta (IIT, Gandhinagar)

When: Thursday, Aug 19, 2021 - 3:55 PM - 5:00 PM (IST)

Abstract:

Nikolai Sergeevich Koshliakov was an outstanding Russian mathematician who made phenomenal contributions to number theory and differential equations. In the aftermath of World War II, he was one among the many scientists who were arrested on fabricated charges and incarcerated. Under extreme hardships while still in prison, Koshliakov (under a different name `N. S. Sergeev') wrote two manuscripts out of which one was lost. Fortunately the second one was published in 1949 although, to the best of our knowledge, no one studied it until the last year when Prof. Atul Dixit and I started examining it in detail. This manuscript contains a complete theory of two interesting generalizations of the Riemann zeta function having their genesis in heat conduction and is truly a masterpiece! In this talk, we will discuss some of the contents of this manuscript and then proceed to give some new results (modular relations) that we have obtained in this theory. This is joint work with Prof. Atul Dixit.

## Thursday, August 5, 2021

### Peter A. Clarkson (University of Kent, UK) - Thursday, Aug 5, 2021 - 3:55 PM - 5:00 PM (IST)

The next speaker in our seminar is Professor Peter Clarkson of the University of Kent, Canterbury, UK.

Here is the announcement.

Talk Announcement:

Title: Special polynomials associated with the Painlevé equations
Speaker: Peter A. Clarkson (University of Kent, UK)

When: Thursday, Aug 5, 2021 - 3:55 PM - 5:00 PM (IST)

Around the middle of the 20th century, as science and engineering continued to expand in new directions, a new class of functions, the Painlevé functions, started to appear in applications. The list of problems now known to be described by the Painlevé equations is large, varied and expanding rapidly. The list includes, at one end, the scattering of neutrons off heavy nuclei, and at the other, the distribution of the zeros of the Riemann-zeta function on the critical line $\mbox{Re}(z) =\tfrac12$. Amongst many others, there is random matrix theory, the asymptotic theory of orthogonal polynomials, self-similar solutions of integrable equations, combinatorial problems such as the longest increasing subsequence problem, tiling problems, multivariate statistics in the important asymptotic regime where the number of variables and the number of samples are comparable and large, and also random growth problems.