Ramanujan's theta functions and sums of triangular numbers

Zhi-Hong Sun

paper

Ramanujan's theta functions and sums of triangular numbers

arXiv:1601.06378

Abstract

Let $Z$ and $N$ be the set of integers and the set of positive integers, respectively. For $a_{1}, a_{2}, \dots, a_{k}, n \in N$ let $N (a_{1}, a_{2}, \dots, a_{k}; n)$ be the number of representations of $n$ by $a_{1} x_{1}^{2} + a_{2} x_{2}^{2} + \dots + a_{k} x_{k}^{2}$ , and let $t (a_{1}, a_{2}, \dots, a_{k}; n)$ be the number of representations of $n$ by $a_{1} \frac{x _{1} ( x _{1} - 1 )}{2} + a_{2} \frac{x _{2} ( x _{2} - 1 )}{2} + \dots + a_{k} \frac{x _{k} ( x _{k} - 1 )}{2}$ $(x_{1}, \dots, x_{k} \in Z$ ). In this paper, by using Ramanujan's theta functions $φ (q)$ and $ψ (q)$ we reveal many relations between $t (a_{1}, a_{2}, \dots, a_{k}; n)$ and $N (a_{1}, a_{2}, \dots, a_{k}; 8 n + a_{1} + \dots + a_{k})$ for $k = 3, 4$ .

33 pages