* Victor Miller <victorsmiller@gmail.com> [Dec 31. 2011 21:36]:
This paper ( http://www.emis.de/journals/HOA/IJMMS/Volume12_2/245.pdf ) looks like something good to read.
Victor
Thanks it's already in my bib. The papers cited in my chapter 31.4 "AGM-type algorithms for hypergeometric functions" (which possibly adds nothing really new) I find quite readable, especially {Robert S.\ Maier: {A generalization of Euler's hypergeometric transform}, % Trans. Amer. Math. Soc. 358 (2006), 39-57. arXiv:math/0302084v4 [math.CA], \bdate{14-March-2006}. URL: \url{http://arxiv.org/abs/math.CA/0302084}.} {Robert S.\ Maier: {Algebraic hypergeometric transformations of modular origin}, arXiv:math/0501425v3 [math.NT], \bdate{24-March-2006}. URL: \url{http://arxiv.org/abs/math.NT/0501425}.} {Robert S.\ Maier: {On rationally parametrized modular equations}, arXiv:math.NT/0611041v4, \bdate{7-July-2008}.} URL: \url{http://arxiv.org/abs/math.NT/0611041}.} {J.\ M.\ Borwein, P.\ B.\ Borwein: {On the Mean Iteration $(a,b)\leftarrow\big(\frac{a+3b}{4},\frac{\sqrt{ab}+b}{2}\big)$}, Mathematics of Computation, vol.53, no.187, pp.311-326, \bdate{July-1989}. J.\ M.\ Borwein, P.\ B.\ Borwein: {A cubic counterpart of Jacobi's Identity and the AGM}, Transactions of the American Mathematical Society, vol.323, no.2, pp.691-701, \bdate{February-1991}. {J.\ M.\ Borwein, P.\ B.\ Borwein, F.\ Garvan: {Hypergeometric Analogues of the Arithmetic-Geometric Mean Iteration}, Constructive Approximation, vol.9, no.4, pp.509-523, \bdate{1993}. URL: \url{http://www.math.ufl.edu/~frank/publist.html}.} {Frank Garvan: {Cubic modular identities of Ramanujan, hypergeometric functions and analogues of the arithmetic-geometric mean iteration}, Contemporary Mathematics, vol.166, pp.245-264, \bdate{1993}. URL: \url{http://www.math.ufl.edu/~frank/publist.html}.} {Kenji Koike, Hironori Shiga: {A three terms Arithmetic-Geometric mean}, Journal of Number Theory, vol.124, pp.123-141, \bdate{2007}.} I didn't cite any paper of Raimundas Vid\={u}nas because all are over my head (as almost all papers that might offer substantial material). cheers, jj