Ternary quadratic forms and linear combination of three triangular numbers

Zhi-Hong Sun

Published 2016-01-24Version 1

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,n\in\Bbb N$ let $N(a,b,c;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2$, and let $t(a,b,c;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 $ $(x,y,z\in\Bbb Z$). In this paper, by using Ramanujan's theta functions we reveal some connections between $t(a,b,c;n)$ and $N(a,b,c;n)$.