Semi-modular forms from Fibonacci-Eisenstein series

Abstract

In recent work, M. Just and the second author defined a class of "semimodular forms" on 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.

On the Pell-Einstein series of power m

“…So it is not surprising that, in a brand new paper in [8], some families called "semi-modular forms" are introduced by Eisenstein-like series that are defined on integer partitions. As a subsequent paper, in [1], a similar problem with Fibonacci number setting is considered. One of the latest works is [19] where Pell-Lucas-Eisenstein Series was defined and gave some properties about them.…”