Semi-modular forms from Fibonacci–Eisenstein series

Abstract

In a 2021 paper, M. Just and the second author defined a class of “semi-modular forms” on $${\mathbb {C}}\backslash {\mathbb {R}}$$ , in analogy with classical modular forms, that are “half modular” in a particular sense; and constructed families of such functions as Eisenstein-like series using symmetries related to integer partitions. Looking for further natural examples of semi-modular behavior, here we construct a family of Eisenstein-like series to produce semi-modular forms, using symmetries related to Fibonacci numbers instead of partitions. We then consider other Lucas sequences that yield semi-modular forms.