## The PIMS Postdoctoral Fellow Seminar: Zafer Selcuk Aygin (Online)

• Date: 11/25/2020
Lecturer(s):

Zafer Selcuk Aygin

Bio: Zafer Selcuk Aygin completed his Ph. D. in mathematics at Carleton University in 2016. He was a research fellow at Nanyang Technological University, Singapore until 2018 and is currently holding a PIMS Post-Doctoral Fellowship at the University of Lethbridge and the University of Calgary on a joint appointment, supported by the Pacific Institute for Mathematical Sciences

Location:

Online

Topic:

Relations between $$\triangle+\cdots + \triangle + 3\triangle+\cdots + 3\triangle$$ and $$\square+\cdots + \square + 3\square+\cdots + 3\square$$

Description:

For non-negative integers $$a,b$$ and $$n$$, let
\begin{align*} t(a,b;n)=& \# \left\{ (x_1,\ldots,x_a,y_1,\ldots, y_b) \in \mathbb{Z}^{a+b} ~ \mid ~n= \frac{x_1(x_1-1)}{2} + \cdots + \frac{x_a(x_a-1)}{2} \right. \\ & \left. + 3\frac{y_1(y_1-1)}{2}+ \cdots + 3\frac{y_b(y_b-1)}{2} \right\} \end{align*}
And
\begin{align*} N(a,b;n)= & \# \{ (x_1,\ldots,x_a,y_1,\ldots, y_b) \in \mathbb{Z}^{a+b} ~\mid~ n=x_1^2 + \cdots + x_a^2 \\ & + 3y_1^2+ \cdots + 3y_b^2 \}. \end{align*}
By works of Bateman and Knopp & Adiga, Cooper and Han, it is known that for $$(a,b) = (1,0), (2,0), (3,0), (4,0), (5,0), (6,0), (7,0)$$, and $$(3,1)$$ we have

\begin{align*} \frac{t(a,b;n)}{N(a,b;8n+a+3b)}=\frac{2}{2^{a+b-2} + 2^{(a+b-2)/2} \cos(\pi (a+3b)/4) +1} \end{align*}

for all $$n \in \mathbb{N}$$. Moreover, for $$(a,b) = (8,0)$$ and $$(2,2)$$, Baruah, Cooper and Hirschhorn has proven that
\begin{align*} \frac{t(a,b;n)}{N(a,b;8n+a+3b)-N(a,b;(8n+a+3b)/4)} = \frac{2}{2^{a+b-2} + 2^{(a+b-2)/2}} \end{align*} for all $$n \in \mathbb{N}$$.

Variations of such identities were observed by many mathematicians including Baruah, Bateman, Cooper, Dastovski, Hirschhorn, Knopp, Sun and Williams. However, it seems that for bigger values of $$a$$ and $$b$$ these ratios start to fluctuate when $$n$$ varies.

In this work we investigate the limiting cases of similar ratios when $$a+b$$ is an even integer greater than 4, $$a \geq 2$$ and $$b \geq 0$$. We show that previously known examples are special cases of an asymptotic relation between $$N(a,b;n)$$ and $$t(a,b;n)$$. We achieve our results by finding certain modular identities which relates generating functions of $$t(a,b;n)$$ to $$N(a,b;n)$$. This is a joint work with Amir Akbary (University of Lethbridge).

This event is part of the Emergent Research: The PIMS Postdoctoral Fellow Colloquium Series.

This event took place via zoom and a recording is available on mathtube.org. To learn about other events in this series and to receive connection details, please register for the event mailing list.

Schedule:

This seminar takes places across multiple timezones: 9:30 AM Pacific/ 10:30 AM Mountain / 11:30 AM Central