Report on joint work with M. Neuhauser. This includes results with C. Kaiser, F. Luca, F. Rupp, R. Troeger, and A. Weisse.

The Lehmer conjecture and Serre’s lacunary theorem describe the vanishing properties of the Fourier coefficients of even powers of the Dedekind eta function.

G.-C. Rota proposed to translate and study problems in number theory and combinatorics to and via properties of polynomials.

We follow G.-C. Rota’s advice. This leads to several new results and improvement of known results. This includes Kostant’s non-vanishing results attached to simple complex Lie algebras, a new non-vanishing zone of the Nekrasov-Okounkov formula (improving a result of G. Han), a new link between generalized Laguerre and Chebyshev polynomials, strictly sign-changes results of reciprocals of the cubic root of Klein’s absolute $j$- invariant, and hence the $j$-invariant itself. Finally we give an interpretation of the first non-sign change of the Ramanujan $\tau(n)$ function by the root distribution of a certain family of polynomials in the spirit of G.-C. Rota.