Victor Miller's cite of PhD thesis by Robert Carls was pretty useless to me -- seems to be a hellish marriage of number theory and topology (??!) without what I wanted, which is complex or real analysis. It seems amazing that they've somehow translated Gauss's AGM into the p-adic number theory world and now use it to do things like count points on finite field versions of "curves"... but at the same time, for mundane analysis purposes this seems useless. However, google then revealed these papers: Takayuki Kato and Keiji Matsumoto: http://www.math.sci.hokudai.ac.jp/~matsu/pdf/FD.pdf we define a (generalized) arithmetic-geometric mean among four terms and express it by Lauricella’s hypergeometric function FD of three variables. Kenji Koike & Hironori Shiga: http://mitizane.ll.chiba-u.jp/metadb/up/irtoroku2/06001.pdf A 3-variable AGM-like iteration yields a certain hyperelliptic integral, also an Appell 2-variable hypergeometric fn. There also are other papers by these same authors. They seem to in some of them be claiming a more general theory. This all seems an impressive breakthrough in analysis which probably will (or should) lead to various new "superfast" algorithms. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)