Crouching AGM, hidden modularity

Title: Crouching AGM, hidden modularity
Creator: Cooper, Shaun; Guillera, Jesus; Straub, Armin; Zudilin, Wadim
Relation: Frontiers In Orthogonal Polynomials and Q-series p. 169-187
Relation: Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes Vol. 1
Publisher Link: http://dx.doi.org/10.1142/9789813228887_0009
Publisher: World Scientific
Resource Type: book chapter
Date: 2018
Description: Special arithmetic series f(x) = ∑∞n=0cn xn, whose coefficients cn are normally given as certain binomial sums, satisfy “self-replicating” functional identities. For example, the equation 1(1 + 4z)2f (z(1 + 4z)3) = 1(1 + 2z)2f (z2(1 + 2z)3) generates a modular form f(x) of weight 2 and level 7, when a related modular parameterization x = x(τ) 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 π and other related constants. Finally, we indicate some possibilities to extend the functional equations to a two-variable setting.
Subject: modular form; arithmetic hypergeometric series; supercongruence; AGM iteration
Identifier: http://hdl.handle.net/1959.13/1420931
Identifier: uon:37655
Identifier: ISBN:9789813228870
Language: eng

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