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.

$$

(n+1)^3A(n+1)=(2n+1)(17n^2+

$$

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})^

\quad

\mbox{and}

\quad

w=q\,\prod_{j=1}^\infty \frac{(1-q^{j})^{12}(1-q^{6j})

$$

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.

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