Universal quaternary mixed sums involving generalized 3-, 4-, 5- and 8-gonal numbers via products of Ramanujan's theta functions

Generalized $m$-gonal numbers are those $p_m(x)=[(m-2)x^2-(m-4)x]\mathrm{/}2$ where $x$ and $m$ are integers with $m\ge 3$. If any nonnegative integer can be written in the form $a{p}_r(h)+b{p}_s(l)+c{p}_t(m)+d{p}_u(n)$, where $a,b,c,d$ are positive integers, then we call $a{p}_r(h)+b{p}_s(l)+c{p}_t(m)+d{p}_u(n)$ a universal quaternary sum. In this paper, we determine the universality of many quaternary sums when $r,s,t,u\in \{3,4,5,8\}$, using the theory of Ramanujan’s theta function identities