On the constant terms of certain meromorphic modular forms for Hecke groups

TL;DR

This paper investigates the constant terms of specific meromorphic modular forms for Hecke groups, exploring their divisibility properties and connections to known integer sequences, with implications for series approximations of pi.

Contribution

It introduces polynomials interpolating these constant terms and links their divisibility properties to well-studied sequences like OEIS A005148.

Findings

Constant terms exhibit notable divisibility properties.

Connections established between modular form constants and OEIS sequences.

Potential applications in series approximations to pi.

Abstract

We study polynomials interpolating the (rational) constant terms of certain meromorphic modular forms for Hecke groups. We make observations about the divisibility properties of the constant terms and connect them to several sequences, for example, to O.E.I.S. sequence A005148 \cite{OEISNewmanShanks}, which was studied by Newman, Shanks and Zagier \cite{newman2004sequence}, \cite{newman2004sequenceAppendix} in an article on its use in series approximations to $\pi$.