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The following infinite Continued Fractions have a couple of characteristics in common.

$2+\cfrac{1^2}{2+\cfrac{2^2}{2+\cfrac{3^2}{2+\cfrac{4^2}{2+\ddots}}}}$

$1+\cfrac{2}{2+\cfrac{4}{3+\cfrac{6}{4+\cfrac{8}{5+\ddots}}}}$

$0+\cfrac{1}{2+\cfrac{3}{4+\cfrac{5}{6+\cfrac{7}{8+\ddots}}}}$

$1+\cfrac{\frac{1}{1}}{1+\cfrac{\frac{1}{3}}{1+\cfrac{\frac{1}{5}}{1+\cfrac{\frac{1}{7}}{1+\ddots}}}}$

$1\cdot 8+\cfrac{1-7^2}{3\cdot 8+\cfrac{1-14^2}{5\cdot 8+\cfrac{1-21^2}{7\cdot 8+\cfrac{1-28^2}{9\cdot 8+\ddots}}}}$

They all are elegant, there is an easy to detect pattern. At the same time, evaluation is for all of them far from trivial. In the last three years I studied many Continued Fractions and I also collected proofs. Many of these proofs were known in the 17th and 18th centuries but have long been hidden in hard-to-access texts. I have summarized my results in a book “Some Continued Fractions and proofs”, free for everyone to download.
Besides revived proofs in modern notation, my book also contains a couple of new and original proofs.

One Continued Fraction in particular still resists any of my attempts to understand its nature: $1^2+\cfrac{1}{2^2+\cfrac{1}{3^2+\cfrac{1}{4^2+\ddots}}}$

I wonder if there is any progress since previous posts.

To quote Jacob Bernoulli: “If anyone finds and communicates to us that which thus far has eluded our efforts, great will be our gratitude”

Paul vdVeen
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  • The book is a jewel. I plotted $f(x)=1^2+x/(2^2+x/(3^2+x/(4^2+\dots)))$. It looks like $\sqrt{x}$ for large positive $x$, but has got funny singularities at some negative points. Around $x_0 \approx -28.3519807194268578227$ it looks like $f(x) \approx -155.566973/(x-x_0)$ and around $x_1 \approx -301.2687320073642607863$ it looks like $f(x) \approx -2967.54234/(x-x_1)$. To be confirmed, as I hope those are not approximation artifacts. – Claude Chaunier Apr 13 '23 at 12:13
  • The next singularity seems to be at $x_2 \approx -1315.072115774073047781$ where $f(x) \approx -18707.6378/(x-x_2)$. – Claude Chaunier Apr 13 '23 at 12:29

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