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

Nasser Abdo Saeed Bulkhali, A. Vanitha, M. P. Chaudhary

Generalized $m$-gonal numbers are those $p_m(x)=[(m-2)x^2-(m-4)x]/2$ where $x$ and $m$ are integers with $m\ge 3$. If any nonnegative integer can be written in the form…