On the reciprocal of Klein's absolute j-invariant and sign changes.

Asai, Kaneko, and Ninomiya utilized properties of Faber polynomials and the denominator formula for the monster Lie algebra to prove that the q-expansion of the reciprocal of Klein's absolute j-invariant has strictly alternating sign changes. In this paper we offer a new and more general approach to study reciprocals. We study families of polynomials Q n (x) attached to normalized arithmetic functions g. They are defined by the q-expansion ∑ n = 0 ∞ Q n (x) q n : = 1 1 - x ∑ n = 1 ∞ g (n) q n . We obtain growth and non-vanishing results. And finally for certain x ∈ R we also obtain statements on the signs of Q n (x) . This includes Weber's γ 2 and Klein's absolute j-invariant. We have actually discovered a formula connecting the polynomials attached to γ 2 and j, which gives a new proof of the result of Asai, Kaneko, and Ninomiya. [ABSTRACT FROM AUTHOR]