Semi-modular forms from Fibonacci-Eisenstein series

TL;DR

This paper introduces a new class of semi-modular forms derived from Fibonacci and Lucas sequences, expanding the understanding of symmetries in Eisenstein-like series beyond classical modular forms.

Contribution

It constructs new semi-modular forms using Fibonacci and Lucas sequence symmetries, broadening the scope of semi-modular form examples beyond previous partition-based methods.

Findings

Fibonacci-based Eisenstein-like series exhibit semi-modular behavior.

Lucas sequences can also generate semi-modular forms.

New examples expand the landscape of semi-modular forms.

Abstract

In recent work, M. Just and the second author defined a class of "semi-modular forms" on $\mathbb C$, 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.