J. Borwein, P. Borwein and F. Garvan; Hypergeometric analogues of the arithmetic-geometric mean iteration , Constr. Approx., 9 (1993), 509-523
Abstract: The arithmetic-geometric mean iteration of Gauss and Legendre is the two-term iteration $a_{n+1}=(a_n+b_n)/2$ and $b_{n+1}=\sqrt{a_n b_n}$ with $a_0:=1$ and $b_0:=x$. The common limit is ${}_2F_{1}(\frac12, \frac12; 1; 1-x^2)^{-1}$ and the convergence is quadratic. This is a rare object with very few close relatives. There are however three other hypergeometric functions for which we expect similar iterations to exist, namely: ${}_2F_{1}(\frac12 - s, \frac12 + s; 1; \cdot)$ with $s=\frac13, \frac14, \frac16$. Our intention is to exhibit explicitly these iterations and some of their generalizations. These iterations exist because of underlying quadratic or cubic transformations of certain hypergeometric functions, and thus the problem may be approached via searching for invariances of the corresponding second-order differential equations. It may also be approached by searching for various quadratic and cubic modular equations for the modular forms that arise on inverting a the ratios of the solutions of these differential equations. In either, case, the problem is intrinsically computational. Indeed, the discovery of the identities and their proofs can be effected almost entirely computationally with the aid of a symbolic manipulation package, and we intend to emphasize this computational approach.