The talk this week is by Pranjal Talukdar of Tezpur university. Here is the announcement of the talk.
Talk Announcement:
Title: On the least $r$-gaps in partitions and identities for the Rogers-Ramanujan Continued fraction
Title: On the least $r$-gaps in partitions and identities for the Rogers-Ramanujan Continued fraction
Speaker: Pranjal Talukdar (Tezpur University)
When: May 22, 2025, 4:00 PM- 5:00 PM IST
Where: Zoom: Please write to the organisers for a link
Best wishes,
Gaurav Bhatnagar, Atul Dixit and Krishnan Rajkumar (organisers)
Abstract:
The minimal excludant of a partition $\pi=(\pi_1,\pi_2,\ldots,\pi_k)$ of $n$ is the smallest positive integer that is not present in $\pi$ and is denoted by $\textup{mex}(\pi)$. The least $r$-gap of $\pi$ is the least positive integer that does not appear in the partition at least $r$ times. In the first half of the talk, we derive some arithmetic functions related to the sum of least $r$-gaps. Using a Tauberian theorem due to Ingham, we obtain Hardy-Ramanujan-type asymptotic formula for two such functions. We also briefly mention some arithmetic properties for these functions.
In the second half, we prove some new identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then
\begin{align*}&R(q)R(q^4)=\dfrac{R(q^5)+R(q^{20})-R(q^5)R(q^{20})}{1+R(q^{5})+R(q^{20})},\\
&\dfrac{1}{R(q^{2})R(q^{3})}+R(q^{2})R(q^{3})= 1+\dfrac{R(q)}{R(q^{6})}+\dfrac{R(q^{6})}{R(q)}.
\end{align*}
The contents of the talk is taken from two chapters of the speaker's Ph.D. thesis completed under the supervision of Prof. Nayandeep Deka Baruah at Tezpur University, India.
