Crouching AGM, Hidden Modularity

TL;DR

This paper explores self-replicating functional equations of special series, their connection to modular forms, and proposes new AGM-type algorithms for computing constants like pi, with potential extensions to two-variable cases.

Contribution

It introduces a novel approach linking self-replicating equations to modular forms and develops AGM-type algorithms for constant computation.

Findings

Identifies functional equations generating modular forms of specific weights and levels.

Proposes new AGM-type algorithms for calculating pi and related constants.

Suggests extensions of these equations to two-variable settings.

Abstract

Special arithmetic series $f(x)=\sum_{n=0}^{\infty}c_nx^n$, whose coefficients $c_n$ are normally given as certain binomial sums, satisfy "self-replicating" functional identities. For example, the equation $$\frac1{(1+4z)^2}f\biggl(\frac{z}{(1+4z)^3}\biggr)=\frac1{(1+2z)^2}f\biggl(\frac{z^2}{(1+2z)^3}\biggr)$$ generates a modular form $f(x)$ of weight 2 and level 7, when a related modular parametrization $x=x(\tau)$ is properly chosen. In this note we investigate the potential of describing modular forms by such self-replicating equations as well as applications of the equations that do not make use of the modularity. In particular, we outline a new recipe of generating AGM-type algorithms for computing $\pi$ and other related constants. Finally, we indicate some possibilities to extend the functional equations to a two-variable setting.