Ramanujan's theta functions and linear combinations of three triangular numbers

Abstract

Let $\Bbb Z$ be the set of integers. For positive integers $a,b,c$ and $n$ 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 $φ(q)$ and $ψ(q)$ we reveal the relation between $t(2,3,3;n)$ and $N(1,3,3;n+1)$, and the relation between $t(1,1,6;n)$ and $N(1,1,3;n+1)$. We also obtain formulas for $t(a,3a,4b;n),$ $t(a,7a,4b;n),t(3a,5a,4b;n)$ and $t(a,15a,4b;n)$ under certain congruence conditions, where $a$ and $b$ are positive odd integers. In addition, we pose many conjectures on $t(a,b,c;n)$ for some special values of $(a,b,c)$.