Relations among representations of integers by certain quadratic forms

Abstract

Let $N(c_1,c_2,\ldots , c_k;n),$ $r(c_1,c_2,\ldots , c_k;n),$ and $t(c_1,c_2,\ldots , c_k;n)$ count the representations of a positive integer n in the quadratic forms

$$\begin{aligned}&c_1\left( x_1^2+x_1 x_2+x_2^2\right) +c_2\left( x_3^2+x_3 x_4+x_4^2\right) +\cdots +c_k\big (x_{2k-1}^2+x_{2k-1}x_{2k}+x_{2k}^2\big ), \\&\quad c_1x_1^2+c_2x_2^2+\cdots +c_kx_k^2, \\&\quad c_1x_1(x_1+1)/2+c_2x_2(x_2+1)/2+\cdots +c_kx_k(x_k+1)/2, \end{aligned}$$

respectively, where $c_i$’s and $x_i$’s are integers. Using Ramanujan’s theta functions $\varphi (q)$, $\psi (q)$, Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by $N(c_1,c_2,\ldots , c_k;n)$, $r(c_1,c_2,\ldots , c_{2k};n)$, and $t(c_1,c_2,\ldots , c_{2k};n)$ when $1<2k \le 8$. We also deduce some new explicit formulas for $r(c_1,c_2,c_3,c_4;n)$ for some particular n when $c_i\in \{1,2,3,6,8,9,16,18,24,48\}.$