## Saturday, March 11, 2023

### Bruce Berndt (UIUC) - Thurs Mar 16 - 6:00 PM (Note Special Time)

Dear all,

We are happy to announce our next talk is by Professor Bruce Berndt, the world's biggest authority on Ramanujan's mathematics and related areas.

Please note the special time. It is two hours later than usual. Please circulate this announcement in your department.

Please see a further announcement below.

Talk Announcement:

Title: Finite Trigonometric Sums: Evaluations, Estimates, Reciprocity Theorems

Speaker: Bruce Berndt (University of Illinois at Urbana Champaign)
When: Mar 16, 2022, 6:00 PM- 7:00 PM IST (7:30 AM - 8:30 AM (CDT))

Abstract
First, motivated by a theorem in Ramanujan's lost notebook, Martino Fassina, Sun Kim, Alexandru Zaharescu, and the speaker developed representations for finite sums of products of trig functions for which we provided theorems and several conjectures.

Second, a paper of Richard McIntosh served as motivation.  First, he made a very interesting conjecture, which was recently proved by Likun Xie, Zaharescu, and the speaker.  Second, he examined a particular trigonometric sum, which inspired Sun Kim, Zaharescu, and the speaker to evaluate in closed form several classes of trigonometric sums, and find reciprocity theorems for others.

New conference announcement
A new conference on algebraic combinatorics has been announced by Arvind Ayyer. It is called
Meru Annual Combinatorics Conference
Dates: 29th to 31st May, 2023

## Sunday, February 26, 2023

### B. Ramakrishnan, ISI, Tezpur - Thursday, Mar 2, 2023 - 4:00 PM

The next talk is by B. Ramakrishnan (popularly known as Ramki), formerly of HRI, Allahabad, and now in ISI, Tezpur.

Talk Announcement:

Title: An extension of Ramanujan-Serre derivative map and some applications.

Speaker: B. Ramakrishnan (Indian Statistical Institute North-East Center, Tezpur)
When: Mar 2, 2022, 4:00 PM- 5:00 PM IST

Abstract

In this talk, we present a simple extension of the Ramanujan-Serre derivative map and
describe how it can be used to derive a general method for explicit evaluation of convolution sums  of the divisor functions. We provide explicit examples for four types of convolution sums.

This is a joint work with Brundaban Sahu and Anup Kumar Singh.

## Saturday, February 11, 2023

### Galina Filipuk, University of Warsaw - Thursday, Feb 16, 2023 - 4:00 PM

We are back to our usual time with a talk by Galina Filipuk all the way from Warsaw, Poland. Please note that we will be open to changing the time, since speakers from the US find this time to be very inconvenient, and we surely would like speakers from the US. The discussions in the previous talk went quite late into the night (for New Zealand) and we thank Shaun Cooper for a very nice talk.

Talk Announcement:

Title: (Quasi)-Painleve equations and Painleve equivalence problem

Speaker: Galina Filipuk (University of Warsaw, Poland)
When: Feb 16, 2022, 4:00 PM- 5:00 PM IST (11:30 CET in Warsaw)

Abstract

Painleve equations are second order nonlinear differential equations solutions of which have no movable critical points (algebraic singularities). They appear in many applications (e.g., in the theory of orthogonal polynomials) but in disguise. How to find a transformation to the canonical form? This is known as the Painleve equivalence problem.
The so-called geometric approach may help in many cases.

In this talk I shall present some recent results on the geometric approach for the Painleve and quasi-Painleve equations.

## Saturday, January 28, 2023

### Ramanujan Special: Shaun Cooper (Massey University) - Thursday, Feb 2, 2023 - 2:30 PM

Happy new year.

The first talk of the year (on Feb 2, 2023) is a "Ramanujan Special". This year's speaker is Shaun Cooper. Please note that the talk will be earlier than usual.

The last year was quite exciting for our group with many talks as well as a mini course. We hope this year is equally exciting. Please consider the seminar to present your latest preprint.

Talk Announcement: Ramanujan Special

Title: Apéry-like sequences defined by four-term recurrence relations: theorems and conjectures

Speaker: Shaun Cooper (Massey University, Auckland, New Zealand)

When: Feb 2, 2022, 2:30 PM- 3:30 PM IST (Note special time) (IST= GMT - 5:30)

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

Abstract

The Apéry numbers are famous for having been introduced and used by R. Apéry to prove that~$\zeta(3)$ is irrational. They may be defined by the recurrence relation
$$(n+1)^3A(n+1)=(2n+1)(17n^2+17n+5)A(n)-n^3A(n-1),$$
with the single initial condition $A(0)=1$ being enough to start the recurrence. The Apéry numbers are all integers, a fact not obvious from the recurrence relation, and they satisfy interesting congruence properties. The generating function
$$y=\sum_{n=0}^\infty A(n)w^n$$
has a splendid parameterisation given by
$$y = \prod_{j=1}^\infty \frac{(1-q^{2j})^7(1-q^{3j})^7}{(1-q^{j})^5(1-q^{6j})^5} \quad \mbox{and} \quad w=q\,\prod_{j=1}^\infty \frac{(1-q^{j})^{12}(1-q^{6j})^{12}}{(1-q^{2j})^{12}(1-q^{3j})^{12}}.$$
In this talk I will briefly survey other sequences defined by three-term recurrence relations that have properties similar to those satisfied by the Apéry numbers described above. I will also introduce some sequences defined by four-term recurrence relations and describe some of their properties.

Several conjectures will be presented.

Here are the slides of the talk.