We know that $$e^{i \pi} = -1 .$$ We can transform all of the components of this identity into (generalized) continued fractions. When we start of with $\pi$, we see that $$ \Big(3+ \mathbf{K}_{i=1}^{\infty} \frac{(\sqrt{a_{i-1}}+2)^2}{6}\Big)^{e \cdot i} = -1 $$ When we take the $\frac{1}{i}$'th power, we see that $$ \Big(3 + \mathbf{K}_{i=1}^{\infty} \frac{(\sqrt{a_{i-1}}+2)^2}{6}\Big)^{e} = (-1)^{-i} ,$$
In which $a_1=1^2,a_2=3^2$. We could re-state this equality by writing $e, -1$ and the root the latter as continued fractions, but I think the question is quite clear without doing so (also: other GCF's of pi). What information about (generalized) continued fractions and calculating with them can we extract from Euler's identity?
Motivation: I was hoping that, if we could find a good explanation for this, we would have more insight in how continued fractions behave under arithmetic operations. With that insight, perhaps we could find ways to combine the continued fractions of certain constants and finding the values of some constants, like Apery's Constant and Catalan's, of which we do know the continued fraction representations but few more.
Thanks,
Max
