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

Abstract

Let $\Bbb Z$ and $\Bbb Z^+$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb Z^+$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2+dw(w+1)/2$ $(x,y,z,w\in\Bbb Z)$. In this paper, by using Ramanujan's theta functions $\varphi(q)$ and $\psi(q)$ we present many formulas and conjectures on $t(a,b,c,d;n)$.