Some relations between t(a,b,c,d;n) and N(a,b,c,d;n)

Abstract

Let $\mathbb{Z}$ and $\mathbb{N}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{N}$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $a{x}^2+b{y}^2+c{z}^2+d{w}^2$, and 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 we reveal some connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$.

19 pages