# Generalizing a property of the complete elliptic integral of the first kind $K(k)$

Define an elliptic modulus $k_n$ such that,

$$\frac{K'(k_n)}{K(k_n)}=\sqrt{n}$$

This is the case $m=2$ of the more general,

$$\frac{\;_2F_1(\frac{1}{m},1-\frac{1}{m},1,1-x)}{\;_2F_1(\frac{1}{m},1-\frac{1}{m},1,x)} = \sqrt{n}$$

where $_2F_1(a,b,c,z)$ is the hypergeometric function, and $\;0< x<1$. For $m = 2,3,4,6$, there are formulas for $x$ in terms of the Dedekind eta function, so I was wondering if there are for other m as well. I found that for $m=5$,

$$\frac{\;_2F_1(\frac{1}{5},\frac{4}{5},1,1-x)}{\;_2F_1(\frac{1}{5},\frac{4}{5},1,x)} = \sqrt{n}$$

let,

$$y = \sqrt{5}\left(\tfrac{1+\sqrt{5}}{2}\right)^\sqrt{5}\exp\left(\tfrac{-\pi\sqrt{n}}{\sin(\pi/5)}\right)$$

then,

$$\frac{25}{x} = \frac{1}{y}+17+38y-\frac{472}{3}y^2+694y^3-\frac{1334744}{375}y^4+\dots$$

Two questions: 1) Is it true that the coefficients of this expansion are all rationals? (If they had been integers, a check with the OEIS would have given a clue.) 2) For $m=5$, can $x$ be also expressed in terms of the Dedekind eta function? I made a related post here more than a week ago, but no one made an answer.

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