Explicit expression for a family of ternary cyclotomic polynomials

In this paper, we give an explicit expression for a certain family of ternary cyclotomic polynomials: specifically ${\mathrm{\Phi }}_{p_1{p}_2{p}_3}$, where $p_1<p_2<p_3$ are odd primes such that $p_2\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}{p}_1$ and $p_3\equiv 1\mathrm{m}\mathrm{o}\mathrm{d}{p}_1{p}_2$. As an application of the explicit expressions, we give an exact formula for the number of nonzero terms in the polynomials in the family, which in turn immediately shows that the density (number of non-zeros terms / degree) is roughly inversely proportional to $p_2$, when $p_1$ is sufficiently large.