TITLE: nth-order generalizations of the arithmetic-geometric mean
QUESTION [15 upvotes]: The arithmetic-geometric mean,
$a_{k+1}=\frac{a_k+b_k}{2} \quad b_{k+1}=\sqrt{a_k b_k}$
is one of the celebrated discoveries of Gauss, who found out that it is equivalent to computing a (complete) elliptic integral (which is a special case of the Gauss hypergeometric function ${}_2 F_1$).
I have been wondering if nth-order generalizations of the iteration,
$a_{k+1}=\frac{a_k+b_k}{n} \quad b_{k+1}=\sqrt[n]{a_k b_k}$
have ever been systematically studied. I've seen this paper by Borwein, but have had trouble searching for other papers. In particular, I'm interested if the coupled sequences also have a common limit, and if so, whether the limit is expressible as a hypergeometric function (or generalizations like those of Appell or Lauricella).
Another possible generalization I thought involves $n$ variables and makes use of the elementary symmetric polynomials. To use $n=4$ as an example:
$a_{k+1}=\frac{a_k+b_k+c_k+d_k}{4}$
$b_{k+1}=\sqrt{\frac{a_k b_k+a_k c_k+a_k d_k+b_k c_k+b_k d_k+c_k d_k}{3}}$
$c_{k+1}=\sqrt[3]{\frac{{a_k b_k c_k}+{a_k b_k d_k}+{a_k c_k d_k}+{b_k c_k d_k}}{2}}$
$d_{k+1}=\sqrt[4]{a_k b_k c_k d_k}$
Would these four sequences (and in general the $n$ sequences) tend to a common limit $F(a_0,b_0,c_0,d_0,\dots)$ like in the $n=2$ case, and if so, are they expressible in terms of known functions?

EDIT
Taking into account Darsh Ranjan's comments, I realized that what I should be looking at instead is the generalization whose denominators are binomial coefficients (thus, the general form $\sqrt[j]{\frac{e_j}{\binom{n}{j}}}$, for $j=1\dots n$ where $e_j$ is the jth elementary symmetric polynomial). The case $n=4$ now looks like
$a_{k+1}=\frac{a_k+b_k+c_k+d_k}{4}$
$b_{k+1}=\sqrt{\frac{a_k b_k+a_k c_k+a_k d_k+b_k c_k+b_k d_k+c_k d_k}{6}}$
$c_{k+1}=\sqrt[3]{\frac{{a_k b_k c_k}+{a_k b_k d_k}+{a_k c_k d_k}+{b_k c_k d_k}}{4}}$
$d_{k+1}=\sqrt[4]{a_k b_k c_k d_k}$
So, still the same question: is there a common limit, and if so, is the limit expressible in terms of known functions?

REPLY [12 votes]: Gauss's hypergeometric formula for the AGM can also be interpreted in terms of a complete elliptic integral
$\int_0^{\pi/2} \phantom. d\theta / \sqrt{a^2 \cos^2 \theta + b^2 \sin^2 \theta}$.  There's a remarkable generalization to complete hyperelliptic integrals in genus 2 arising from a four-variable AGM:
$$
(a,b,c,d) \mapsto \left(
  \frac14(a+b+c+d),
\frac12\bigl(\sqrt{ab}+\sqrt{cd}\phantom.\bigr),
\frac12\bigl(\sqrt{ac}+\sqrt{bd}\phantom.\bigr),
\frac12\bigl(\sqrt{ad}+\sqrt{bc}\phantom.\bigr)
\right)
$$
(whose specialization $a=b$, $c=d$ recovers the usual AGM).  One source available online is

Jarvis, Frazer: Higher genus arithmetic-geometric means, Ramanujan J. 17 (2008), 1–17.