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))

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

Live Link: https://youtu.be/0hPgarkEXdc

Tea or Coffee: Please bring your own.

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)

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.

The
six Painlevé equations, whose solutions are called the Painlevé
transcendents, were derived by Painlevé and his colleagues in the
late 19th and early 20th centuries in a classification of second order
ordinary differential equations whose solutions have no movable critical
points. In the 18th and 19th centuries, the classical special
functions such as Bessel, Airy, Legendre and hypergeometric functions,
were recognized and developed in response to the problems of the day in
electromagnetism, acoustics, hydrodynamics, elasticity and many other
areas. 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.

The
Painlevé equations possess a plethora of interesting properties
including a Hamiltonian structure and associated isomonodromy problems,
which express the Painlevé equations as the compatibility condition
of two linear systems. Solutions of the Painlevé equations have some
interesting asymptotics which are useful in applications. They possess
hierarchies of rational solutions and one-parameter families of
solutions expressible in terms of the classical special functions, for
special values of the parameters. Further the Painlevé equations
admit symmetries under affine Weyl groups which are related to the
associated Bäcklund transformations.

In this talk I shall
discuss special polynomials associated with rational solutions of
Painlevé equations. Although the general solutions of the six
Painlevé equations are transcendental, all except the first
Painlevé equation possess rational solutions for certain values of
the parameters. These solutions are expressed in terms of special
polynomials. The roots of these special polynomials are highly symmetric
in the complex plane and speculated to be of interest to number
theorists. The polynomials arise in applications such as random matrix
theory, vortex dynamics, in supersymmetric quantum mechanics, as
coefficients of recurrence relations for semi-classical orthogonal
polynomials and are examples of exceptional orthogonal polynomials.