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

Zhi-Hong Sun

Published 2015-11-19Version 1

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\in\Bbb Z$). In this paper we reveal some connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\in\Bbb N$ and $a\equiv 1\pmod 2$. We show that $$t(a,b,c,d;n)=\frac 23(N(a,b,c,d;8n+a+b+c+d)-N(a,b,c,d;2n+(a+b+c+d)/4))$$ for $(a,b,c,d)= (a,a,2a,8m+4)$ and $(a,3a,4k+2,4m+2)$ with $k\equiv m\pmod 2$.