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

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.

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