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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]

This paper develops a general abstract non-holomorphic operator calculus under minimal regularity requirements on the family of operators through the concept of algebraic eigenvalue and the use of a, very recent, transversalization theory. Further, it analyzes under what conditions the inverse of a non-analytic family admits a finite Laurent development, and employs the new findings to calculate the multiplicity of a real non-analytic family through a logarithmic residue, so extending the applicability of the classical theory of I. C. Gohberg and coworkers. Applications to matrix families and Nonlinear Analysis are also explained. [ABSTRACT FROM AUTHOR]