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

Abstract

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

arXiv comment35 pages