On the number of representations of n as a linear combination of four triangular numbers II

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 the connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\in \mathbb{N}$ and $2\nmid a$. We show that $$t(a,b,c,d;n)=\frac{2}{3}N(a,b,c,d;8n+a+b+c+d)-2N(a,b,c,d;2n+(a+b+c+d)/4)$$ for $(a,b,c,d)=(a,a,2a,8m),(a,3a,8k+2,8m+6),(a,3a,8m+4,8m+4)(n\equiv m+\frac{a-1}{2}(\mathrm{mod}2))$ and $(a,3a,16k+4,16m+4)(n\equiv \frac{a-1}{2}(\mathrm{mod}2))$. We also obtain explicit formulas for $t(a,b,c,d;n)$ in the cases $(a,b,c,d)=(1,1,2,8),(1,1,2,16),(1,2,3,6),(1,3,4,12),(1,1,$ $3,4),(1,1,5,5),(1,5,5,5),(1,3,3,12),(1,1,1,12),(1,1,3,12)$ and $(1,3,3,4)$.

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