Olivia Yaoxiang Mei, Fanghou Qing, Fan Yan, Zhang Yan: Some new relations between sums of squares and triangular numbers, 305-315

Abstract:

Let $T(a_1,a_2,a_3,a_4;n)$ denote the number of representations of $n$ as $a_1\frac{x_1(x_1+1)}{2}+ a_2\frac{x_2(x_2+1)}{2}+a_3\frac{x_3(x_3+1)}{2} +a_4\frac{x_4(x_4+1) }{2}$, where $a_1,a_2,a_3 , a_4$ are positive integers, $n$, $x_1$, $x_2$, $x_3$, $x_4$ are arbitrary nonnegative integers, and let $N(a_1,a_2,a_3,a_4;n)$ denote the number of representations of $n$ as $a_1x_1^2+a_2x_2^2+a_3x_3^2 +a_4x_4^2$, where this time $x_1,x_2,x_3,x_4$ are integers. In a recent paper, Sun not only discovered many relations between $T(a_1,a_2,a_3,a_4;n)$ and $N(a_1,a_2,a_3,a_4;n)$, but also posed a number of conjectures on the relations between $T(a_1,a_2,a_3,a_4;n)$ and $N(a_1,a_2,a_3,a_4;n)$. In this paper, we confirm some of Sun's conjectures by using Ramanujan's theta function identities and $(p, k)$-parametrization of theta functions.

Key Words: Ramanujan's theta function identities, sum of squares, sum of triangular numbers.

2010 Mathematics Subject Classification: Primary 11D85; Secondary 11E25.