## Crouching AGM, hidden modularity.(English)Zbl 1437.11058

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
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