1
$\begingroup$

$\gamma$, the Euler-Mascheroni constant, has the following simple regular continued fraction:

$$\gamma=[0; 1, 1, 2, 1, 2, 1,\dots]=0+ \cfrac{1}{ 1+\cfrac{1}{ 1+\cfrac{1}{ 2+\cfrac{1}{ 1+\ddots } } } }$$

I wondered which would be its negative continued fraction, but after searching I have not found anything about it in the literature I have checked.

Just using rough approximation methods, I have derived that it seems that

$$\gamma=1- \cfrac{1}{ 3-\cfrac{1}{ 2-\cfrac{1}{ 3-\cfrac{1}{ 2-\ddots } } } }$$

(although there is an initial pattern in the alternating $2$ and $3$, this pattern ceases).

I would like to know if there is some way to transform a regular continued fraction into a negative continued fraction, to see if that transformation could give some valuable info about $\gamma$. Also, if the negative continued fraction of $\gamma$ is already known, it would be great if you could provide where I can consult it.

Thanks in advance!

$\endgroup$
1
$\begingroup$

I found this interesting research that I share with you in case you are interested in the relationship between the negative and positive continued fractions:

https://scholarship.claremont.edu/cgi/viewcontent.cgi?filename=2&article=1183&context=hmc_theses&type=additional

According to Definition 8, the first terms of the negative continued fraction of the Euler Mascheroni constant would be:

$$\gamma=[1,3,2,3,2,3,2,2,2,5,2,2,2,2,2,2,2,2,2,2,2,2,7,3,2,2,2,2,2,2,2,3,2,...]$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.