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.

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