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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]

Singularity analysis of ordinary differential equations is an important tool in the determination of the possible integrability of the equations. Although singularity analysis has been studied for decades, it still seems to cause problems in terms of the interpretation of some elements of the analysis. We summarise these problems and try to present a delicate approach towards their resolution by separating the treatment for autonomous and nonautonomous equations. [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]

This paper is the sequel to the paper “Commutative algebras of rational function matrices as endomorphisms of Kronecker modules I”. Here we focus on those Kronecker modules that are rank-2 extensions of finite-dimensional, torsion-free modules by the unique generic module R. We show that if such an extension is indecomposable, then its endomorphism algebra is trivial. This resolves the missing case from the previous paper, where the effects of height functions on the endomorphism algebras of extensions are studied. [Copyright &y& Elsevier]