Can I find iterated Root Mean Square-Arithmetic Mean as a function of Arithmetic-geometric mean (AGM) with some transformations if it is possible?
if not possible, what is the closed form of it as known functions ?
$$AGM=M(x,y)=\frac{\pi}{4}\frac{x+y}{K(\frac{x-y}{x+y})}$$
where $K(m)$ is the complete elliptic integral of the first kind:
$$K(m)=\int_{0}^{\frac{\pi}{2}} \frac{dx}{\sqrt{1-m^2\sin^2(x)}}$$
Iterative Root-Mean Square-Arithmetic Mean calculation:
$$r_1=\sqrt{\frac{r_0^2+a_0^2}{2}}$$
$$a_1=\frac{r_0+a_0}{2}$$
$$r_{n+1}=\sqrt{\frac{r_n^2+a_n^2}{2}}$$
$$a_{n+1}=\frac{r_n+a_n}{2}$$
Root Mean Square-Arithmetic Mean of $(r_0,a_0)=RMSAM(r_0,a_0)=\lim\limits_{n\to \infty} r_{n}=\lim\limits_{n\to \infty} a_{n}$
Thanks a lot for answers