Define the two similar continued fractions,
$$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$
$$y=\cfrac{1}{km\color{blue}-\cfrac{(m-1)(m+1)} {3km\color{blue}-\cfrac{(2m-1)(2m+1)}{5km\color{blue}-\cfrac{(3m-1)(3m+1)}{7km\color{blue}-\ddots}}}}\tag2$$
For integer $m>1$, it seems $(1)$ and $(2)$ are algebraic numbers. For $(1)$ and the special case $k=1$, then,
$$x=\tan\Big(\frac{\pi}{4m}\Big)=\frac{1}{4m}\frac{\Gamma\Big(\frac{2m-1}{4m}\Big)\Gamma\Big(\frac{2m+1}{4m}\Big)}{\Gamma\Big(\frac{4m-1}{4m}\Big)\Gamma\Big(\frac{4m+1}{4m}\Big)}\tag3$$
and is adapted from Nicco's cfrac in this post. For $(2)$ and general $k\neq1$,
$$y =\tanh\Big(\frac{1}{2m}\,\log (R)\Big)= \frac{R^{1/m}-1}{R^{1/m}+1};\;\;R=\frac{k+1}{k-1}\tag4$$
and solves,
$$\frac{ (1+y)^{m}-(1-y)^{m}}{(1+y)^{m}+(1-y)^{m}}=\frac{1}{k}\tag5$$
hence $x,y$ are radicals.
Questions:
- What is the formula for $x$ for general $k>1$?
- Does $y$ also have a formula in terms of gamma functions?