How find this Continued fraction $[1,3,5,7,9,11,\cdots]$ value.

Question

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

Answer 1

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.

Answer 2

According to The WOLFRAM Functions Site, that is the continued fraction of $ \alpha = \coth 1 = (e^2+1)/(e^2-1)$.

The continued fraction for $\coth z$ (with proof) can be found in the Handbook of Continued Fractions for Special Functions.