Universal sums via products of Ramanujan's theta functions

An integer-valued polynomial $P(x,y,z)$ is said to be universal (over $\mathbb{Z}$) if each nonnegative integer can be written as $P(x,y,z)$ with $x,y,z\in \mathbb{Z}$. In this paper, we mainly introduce a new technique to determine the universality of some sums in the form $x(a_{1}x+a_2)\mathrm{/}2+y(b_{1}y+b_2)\mathrm{/}2+z(c_{1}z+c_2)\mathrm{/}2$ (with $a_1-a_2,b_1-b_2,c_1-c_2$ all even) conjectured by Sun, using various identities of Ramanujan’s theta functions. For example, we prove that $x(3x+1)+y(3y+2)+2z(3z+2)$ and $x(4x+r)+y(3y+1)\mathrm{/}2+z(7z+1)\mathrm{/}2(r=1,3)$ are universal.