The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad a_{n+1}=\frac{a_n+b_n}2,\quad b_{n+1}=\sqrt{a_n b_n}.$$ $$\text{Then}\quad\operatorname{AGM}(a,b)=\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n.$$ The arithmetic-geometric mean can be expressed in a closed form using the complete elliptic integral of the first kind and elementary functions.

Let us try to generalize the arithmetic-geometric mean to 3 numbers $a,b$ and $c$. One way to define it would be just as $\operatorname{AGM}\left(\frac{a+b+c}3,\sqrt[3]{abc}\right)$. Apparently, this gives us nothing really new or interesting.

Let us consider a different approach: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad c_0=c,$$ $$\quad a_{n+1}=\operatorname{AGM}(b_n,c_n),\quad b_{n+1}=\operatorname{AGM}(a_n,c_n),\quad c_{n+1}=\operatorname{AGM}(a_n,b_n).$$ $$\text{Then}\quad\operatorname{AGM}(a,b,c)=\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n=\lim_{n\to\infty}c_n.$$ This gives us a function different than one in the previous approach. For example, we can calculate that $$\operatorname{AGM}(1,2,3)\approx1.909157449373156462538798818255615478726285889167...$$ (you can see more digits here)

Have this function and its properties been already studied? What is known about it? Is it possible to express $\operatorname{AGM}(a,b,c)$ (or, at least, some of its non-trivial special values) in a closed form using known special functions?

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    $\begingroup$ I assume this is related to nesting complete elliptic integrals but I couldn't find anything on the subject. Elliptic rational functions was the closest thing I could find but I think they are still too different. $\endgroup$ – Ali Caglayan May 22 '16 at 1:22
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    $\begingroup$ There is a generalization of AGM (see the cubic version at mathoverflow.net/q/202008/15540) but this one for three numbers is new. +1 for the same. $\endgroup$ – Paramanand Singh May 22 '16 at 11:41
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    $\begingroup$ Could this be the same limit? math.stackexchange.com/questions/442062/… $\endgroup$ – alphacapture May 24 '16 at 19:26
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    $\begingroup$ @alphacapture, no, it's not the same. For the question you referenced the limit for $1,2,3$ will be $1.9099262335408153237$ $\endgroup$ – Yuriy S Aug 18 '16 at 18:16
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    $\begingroup$ Have you tried $(1,1,\sqrt{2})$ or $(1,\sqrt{2},\sqrt{2})$ since the only closed form for a classical AGM appears for $1$ and $\sqrt{2}$? $\endgroup$ – Yuriy S Aug 23 '16 at 12:25

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