TITLE: How find this Continued fraction $[1,3,5,7,9,11,\cdots]$ value.
QUESTION [5 upvotes]: show this:
$$\alpha=[1,3,5,7,9,11,\cdots]=1+\dfrac{1}{3+\dfrac{1}{5+\dfrac{1}{7+\dfrac{1}{\cdots}}}}=\dfrac{e^2+1}{e^2-1}$$
I found wiki Continued fraction also not have this problem,maybe this problem can't have simple closed form?
Thank you for you give someusefull

REPLY [6 votes]: The entire function $g(z)=\frac{\sinh z}{z}$ satisfies the differential equation $(z g)''=z g$, or:
$$ zg''+2g' = zg,\tag{1}$$
or:
$$ \frac{g'}{g}=\frac{1}{\frac{2}{z}+\frac{g''}{g'}}.\tag{2}$$
By differentiating $(1)$, we get similar expressions for $\frac{g^{(n+1)}}{g^{n}}$, hence a continued fration representation for:
$$ \frac{g'}{g}=\frac{d}{dz}\log g = \coth z - \frac{1}{z}, $$
as wanted.