Crouching AGM, Hidden Modularity

Abstract

Special arithmetic series f (x) = ∞ n=0 c n x n , whose coefficients c n are normally given as certain binomial sums, satisfy 'self-replicating' functional identities. For example, the equationgenerates a modular form f (x) of weight 2 and level 7, when a related modular parametrization x = x(τ ) 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 π and other related constants. Finally, we indicate some possibilities to extend the functional equations to a two-variable setting.

Hypergeometric Modular Equations

“…Finally, we notice that the identities in Theorems 3.1, 4.1 and 5.1 can be used in designing AGM-type algorithms [7] for the effective computation of π and other mathematical constants. The details of such applications can be found in [16,19].…”

Self-replication and Borwein-like algorithms

“…The point of view of the Borweins is explained in [7], and their proofs require a good knowledge of the elliptic modular functions and forms. In [11] we presented a different and simple strategy and we used it further in [9] applying it to new self-replicating identities of non-hypergeometric type. It is precisely in [9] where we named the technique as selfreplication method, and its property as Hidden Modularity, which makes knowledge of modularity (although interesting) unnecessary to prove the algorithms.…”

“…In [11] we presented a different and simple strategy and we used it further in [9] applying it to new self-replicating identities of non-hypergeometric type. It is precisely in [9] where we named the technique as selfreplication method, and its property as Hidden Modularity, which makes knowledge of modularity (although interesting) unnecessary to prove the algorithms. In this paper we generalize, with a free parameter, some Borwein algorithms for the number π, to include some values of the Gamma function, like Γ(1/3), Γ(1/4) and of course Γ(1/2) = √ π.…”