On the Lacunarity of some eta-products

TL;DR

This paper investigates the lacunarity of certain eta-products, establishing precise conditions for lacunarity and expressing these series as linear combinations of CM forms.

Contribution

It characterizes when specific eta-products are lacunary and provides their representation as linear combinations of CM forms, advancing understanding of their structure.

Findings

Eta-products are lacunary if and only if b is 1, 2, 3, 4, or 16.

Lacunary eta-products can be expressed as linear combinations of CM forms.

The paper provides a complete characterization of lacunarity for the considered eta-products.

Abstract

The lacunarity is an interesting property of a formal series. We say a series is lacunary if "almost all" of its coefficients are zero. In this article we considered about the lacunarity of some eta-products like \eta(z)^2\eta(bz)^2, and proved that they are lacunary if and only if b is 1,2,3,4 or 16. Then We write them as linear combinations of some CM forms.