On Zagier's missing continued fraction with multiple limits?

Question

I. Zagier's continued fraction

As pointed out by Gorodetsky in his answer, Zagier evaluated the continued fractions associated with his six sporadic sequences excepting the one for $(-9,-3,-27)$. Let $\color{red}{c=-27,}$ then,

$$C_2=\cfrac{1}{-3 + \cfrac{1^4\,c}{-21 + \cfrac{2^4\, c}{-57+ \cfrac{3^4\,c}{-111 +\ddots }}}}$$

or more compactly,

$$C_2(n) = \frac1{-3 + \large{\underset{k=1}{\overset{n}{\mathrm K}} ~ \frac{-27k^4}{-(9k^2+9k+3)}}}$$

---

II. Multiple Limits

From the paper "Continued Fractions with Multiple Limits", it turns out a cfrac can have multiple limits, a famous one due to Ramanujan (but of course) with two limits depending on its odd or even approximants.

Zagier's cfrac $C_2(n)$ seems to have six limits, based on approximants $\text{mod}\; 6,$

\begin{align} \lim_{m\to\infty} C_2(6m+0)& \overset{\color{red}?}= -0.3906\dots = -\frac{2}{3\sqrt3}\kappa\\ \lim_{m\to\infty} C_2(6m+1)& \overset{\color{red}?}= -0.6343\dots\\ \lim_{m\to\infty} C_2(6m+2)& \overset{\color{red}?}= -1.1217\dots\\ \lim_{m\to\infty} C_2(6m+3)& = \;\;divergent\\ \lim_{m\to\infty} C_2(6m+4)& \overset{\color{red}?}= +0.3404\dots\\ \lim_{m\to\infty} C_2(6m+5)& \overset{\color{red}?}= -0.1469\dots\\ \end{align}

The trend is more visible in the table below:

$$\begin{array}{|c|c|c|c|c|c|} \hline m&0&1&2&3&4&5\\ \hline 5000&-0.3906430& -0.634340& -1.121715& >10^4&0.340448& -0.146952\\ \hline 16666&-0.3906499& -0.634343& -1.121728& >10^5& 0.340435& -0.146955\\ \hline 50000&\color{blue}{-0.3906508}& -0.634344& -1.121731& >>10^5& 0.340432& -0.146956\\ \hline \end{array}$$

Excepting $C_2(6m+3)$, they seem to be converging to certain values as $m$ increases, though very slowly. (My thanks to user Domen from the Mathematica SE for extending the table two more layers.) The only one with an apparent closed-form is the first,

$$-\frac{2}{3\sqrt3}\kappa = \color{blue}{-0.3906512}$$

where $\kappa = \operatorname{Cl}_2\left(\tfrac13\pi\right)$ is Gieseking's constant (which also appears in Zagier's other cfracs.)

---

III. Question

1. With one obvious exception, are the rest actually converging to something?
2. If so, do they have closed-forms and is the first guess correct?

---

P.S. In Mathematica, the command is,

N[1/(-3 + ContinuedFractionK[-27k^4,-(9k^2+9k+3), {k, 1, n}]),20]

Answer 1

Set $Q=(1/2)L(\chi_{-3},2)$ (related to your Gieseking constant) and $P=2\pi^2/81$. The limits are almost certainly (not proved),

\begin{align} \lim_{m\to\infty}C_2(6m+0) &= -Q\\ \lim_{m\to\infty}C_2(6m+1) &= -P-Q\\ \lim_{m\to\infty}C_2(6m+2) &= -3P - Q\\ \lim_{m\to\infty}C_2(6m+3) &= \infty\\ \lim_{m\to\infty}C_2(6m-2) &= 3P - Q\\ \lim_{m\to\infty}C_2(6m-1) &= P - Q \end{align}

Remark: I use a powerful extrapolation method explained for instance in a recent book of mine with K. Belabas, and in a few seconds obtain the limits to 38 decimals, which allowed me to guess the limits.

Answer 2

To complete the 12 cfracs in this post and the 4 in the next, all associated with 16 "sporadic sequences", then 13 of them have closed-forms, 1 has six limits (also with closed-forms but one divergent), and the last 2 as divergent.

We evaluate the last 3 cfracs for $n = 1000\; \text{to}\; 1200,$ sort the values $v$, and plot the values $-2<v<2.$

---

I. Degree 2 for (-9,-3,-27)

$$C_2(-9,-3,-27) = \frac1{-3 + \large{\underset{k=1}{\overset{n}{\mathrm K}} ~ \frac{-27k^4}{(-9k^2-9k-3)}}}$$

As discussed in the post above, this has six limits as clearly seen in the plot below (the divergent $v \to \infty$ is not shown),

---

II. Degree 3 for (11,3,1)

$$C_3(11,3,1) = \frac1{-5 + \large{\underset{k=1}{\overset{n}{\mathrm K}} ~ \frac{-125k^4}{-(2k+1)(11k^2+11k+5)}}}$$

Its plot is vastly different and presumably has infinitely many limits within $-1<v<0$,

It has an illusory "pattern" if $n$ is mod $5$ or mod $7$ which misled me for a while, but it disappeared with further analysis.

---

III. Degree 3 for (7,2,8)

$$C_3(7,2,8) = \frac1{-3 + \large{\underset{k=1}{\overset{n}{\mathrm K}} ~ \frac{-81k^4}{-(2k+1)(7k^2+7k+3)}}}$$

Likewise, its plot looks very similar to the previous one,

so the same conclusion about it can be made.