Crouching AGM, Hidden Modularity

Special arithmetic series $f(x)={\sum }_{n=0}^{\mathrm{\infty }}{c}_n{x}^n$, whose coefficients $c_n$ are normally given as certain binomial sums, satisfy “self-replicating” functional identities. For example, the equation $$\frac{1}{(1+4z{)}^2}f\left(\frac{z}{(1+4z{)}^3}\right)=\frac{1}{(1+2z{)}^2}f\left(\frac{z^2}{(1+2z{)}^3}\right)$$ 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.