Yes, i used unclear notation.

At first, i was studying some orthogonal function (in control theory) and more specifically its sign. The orthogonal function depends of four parameters and its linked with a polynomial (such as the one of the question). After a big number of numerical tests, i observe that when i take the parameters in the form $(abc,abd,acd,bcd)$ the polynomial always has a negative coefficient (that is important to understand the orthogonal function). I don't know if it's clear, it is always complicated to summarize a problem.

When we write $P(X)=\prod \phi_{m}(X)$ it's not obvious for me that we can deduce the sign of the coefficient of $P$. For exemple if we consider $$Q(X)=\frac{(1-X^{ab})(1-X^{ac})(1-X^{bc})}{(1-X^{a})(1-X^{b})(1-X^{c})}$$ we have also $Q(X)=\prod \phi_{m}(X)$ but in this case it's clear that $Q$ only has non negative coefficient.

@Louis the references you give seem really interesting...