Crouching AGM, hidden modularity. (English) Zbl 1437.11058

Cooper, Shaun; Guillera, Jesús; Straub, Armin; Zudilin, Wadim

Nashed, M. Zuhair (ed.) et al., Frontiers in orthogonal polynomials and \(q\)-series. Papers based on the international conference, University of Central Florida, Orlando, FL, USA, May 10–12, 2015. Dedicated to Professor Mourad Ismail on his 70th birthday. Hackensack, NJ: World Scientific. Contemp. Math. Appl., Monogr. Expo. Lect. Notes 1, 169-187 (2018).

Summary: Special arithmetic series \(f(x) = \sum_{n=0}^{\infty} c_n x^n\), 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 parameterization \(x=x(\tau)\) is properly chosen. In this chapter 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.
For the entire collection see [Zbl 1407.33001].

MSC:

11F11	Holomorphic modular forms of integral weight
33C20	Generalized hypergeometric series, \({}_pF_q\)
33F05	Numerical approximation and evaluation of special functions
65Q30	Numerical aspects of recurrence relations

Keywords:

modular form; arithmetic hypergeometric series; supercongruence; identity for \(1/\pi\); AGM iteration