I. Some functions
As these will be used in the continued fraction evaluations below, recall the Riemann zeta function $\zeta(s),$ and Dirichlet beta function $\beta(s),$
$$\beta(s) = \sum_{n=1}^\infty\frac{(-1)^{n-1}}{(2n-1)^s}$$
and special cases of the Clausen function $\operatorname{Cl}_s(x),$
$$\operatorname{Cl}_2(x) = \sum_{n=1}^\infty\frac{\sin(n\,x)}{n^2}$$
\begin{align}
\operatorname{Cl}_2\left(\tfrac12\pi\right) &= K = \beta(2) \\
\operatorname{Cl}_2\left(\tfrac13\pi\right) &= \kappa
\end{align}
with Catalan's constant $K$ and its cubic counterpart Gieseking's constant $\kappa$.
II. Zagier's 6 sporadic sequences
Inspired by Apery's result in proving the irrationality of $\zeta(3)$ using certain integer sequences, Zagier (via a computer) searched for sequences with recurrence relation and deg-$2$ coefficients of form,
$$(n+1)^2\,u_{n+1} = (an^2+an+b)u_k+ cn^2\,u_{n-1}$$
that produced only integer values. Only six $(a,b,c)$ were found, namely,
$$(11,3,1),\quad (7,2,8) ,\quad (12,4,-32)$$
$$(-17,-6,-72),\quad (10,3,-9), \quad (-9,-3,-27)$$
It seems we can use ALL these coefficients to produce nice cfracs.
III. Continued fractions
Given a 3-term recurrence relation of form,
$$F_1(n)\,u_{n+1} = F_2(n)\,u_n + F_3(n)\,u_{n-1}$$
where $F_i(n)$ are polynomials of degree $k$. Define two polynomial functions using the rules,
\begin{align}
p(n) &= F_1(n-1)\, F_3(n)\\
q(n) &= F_2(n)
\end{align}
which implies $p(n)$ has degree twice that of $q(n)$. Define the continued fraction,
$$C =\cfrac{1}{q(0) + \cfrac{p(1)}{q(1) + \cfrac{p(2)}{q(2)+ \cfrac{p(3)}{q(3)+\ddots } }}}$$
More compactly,
$$C(m) = \frac1{q(0) + \large{\underset{n=1}{\overset{m}{\mathrm K}} ~ \frac{p(n)}{q(n)}}}$$
or in Mathematica notation,
$$C(m) = \frac1{q(0) + \text{ContinuedFractionK}[p(n),\;q(n),\, \text{{n, 1, m}}]}$$
It seems $C$ may have a nice closed-form based on the properties of the recurrence relation. Examples below.
IV. Degree 2
Recall Zagier's recurrence,
$$\color{blue}{(n+1)^2}\,u_{n+1} = (\color{blue}{an^2+an+b})u_k+\color{blue}{cn^2}\,u_{n-1}$$
Define $p(n)$ and $q(n)$ according to the rules in the previous section,
\begin{align}
p(n) &= \color{blue}{n^2}\times \color{blue}{cn^2} = cn^4\\
q(n) &= \color{blue}{an^2+an+b}
\end{align}
Then define the cfrac,
$$C_2(a,b,c) = \frac1{q(0) + \large{\underset{n=1}{\overset{\infty}{\mathrm K}} ~ \frac{p(n)}{q(n)}}}$$
Q: Is it true that,
\begin{align}
C_2(11,3,1) &= \frac15\,\zeta(2)\\
C_2(-17,-6,-72) &=\color{green}{-\frac5{6\sqrt3}\operatorname{Cl}_2\left(\tfrac13\pi\right) = -\frac5{6\sqrt3}\kappa}\\
C_2(10,3,-9) &=\frac2{3\sqrt3}\operatorname{Cl}_2\left(\tfrac13\pi\right) = \frac2{3\sqrt3}\kappa\\
C_2(7,2,8) &= \frac14\,\zeta(2)\\
C_2(12,4,-32) &= \frac12\operatorname{Cl}_2\left(\tfrac12\pi\right) = \frac12\beta(2)=\frac12K\\
C_2(-9,-3,-27) &=\;\color{red}{??}
\end{align}
where $K$ is Catalan's constant and $\kappa$ is Gieseking's constant, both of which not yet proven to be irrational.
Note: The first evaluation is valid since it was found by Apery, while the second (in $\color{green}{\text{green}}$) is courtesy of H. Cohen's answer. (Update: May 22, 2023) It turns out $C_2(-9,-3,-27)$ has six limits, one of which is divergent. See this MO post.
V. Degree 3
In Cooper's paper, we find the recurrence relation with deg-$3$ coefficients in $n$,
$$(n+1)^3\,v_{n+1} = -(2n+1)(an^2+an+a-2b)v_n +(-a^2-4c)n^3\,v_{n-1}$$
and Zagier's $(a,b,c).$ Using the same rules, let,
\begin{align}
r(n) &= n^3\times(-a^2-4c)n^3 = -(a^2+4c)n^6\\
s(n) &= -(2n+1)(an^2+an+a-2b)
\end{align}
Define the cfrac,
$$C_3(a,b,c) = \frac1{s(0) + \large{\underset{n=1}{\overset{\infty}{\mathrm K}} ~ \frac{r(n)}{s(n)}}}$$
Q: Is it true that,
\begin{align}
C_3(11,3,1) &=\;\color{red}{??}\\
C_3(-17,-6,-72) &= \frac16\,\zeta(3)\\
C_3(10,3,-9) &= -\frac{7}{24}\,\zeta(3)\\
C_3(7,2,8) &=\;\color{red}{??}\\
C_3(12,4,-32) &= -\frac{7}{32}\,\zeta(3)\\
C_3(-9,-3,-27) &= \frac{128}{243\sqrt3}\,\beta(3) = \frac{4\pi^3}{243\sqrt3}
\end{align}
where $d = a^2+4c =125, 1, 64, 81, 16, -27,$ respectively (and all powers of the smallest primes $2,3,5$).
Note: The second closed-form is valid since it was also found by Apery which he used (together with other methods) to prove the irrationality of $\zeta(3)$.
VI. Degree 4 & 5
Curiously, there is no known 3-term recurrence,
$$P_1(n) v_{n+1} = P_2(n) v_n + P_3(n) v_{n-1}$$
where $P_i$ are polynomials of deg-$4$. Why? But Zudilin found,
$$Q_1(n) v_{n+1} = Q_2(n) v_n + Q_3(n) v_{n-1}$$
where $Q_i$ are polynomials of deg-$5$ and used it in an analogous continued fraction for $\zeta(4).$ (To be discussed in the next post.)
VII. Questions
- Are all cfracs with proposed closed-forms correct? (I know two of them are.)
- What are the closed-forms of the others?
Best Answer
We have $$C_2(-17,-6,-72)=-(5/8)L(\chi_{-3},2)$$ and $$C_2(10,3,-9)=(1/2)L(\chi_{-3},2)$$ so both are proportional to what you call Gieseking's constant but which is simply the value at 2 of the L function of the nontrivial character modulo 3, close analogue to Catalan's constant which is the same with the nontrivial character modulo 4.
All the other "??" that you quote, both in degree 2 and in degree 3 are divergent cfracs (by the way, "degree" is more proper than "level").
Finally just a typo: $C_3(12,4,-32)=-(7/32)\zeta(3)$ (minus sign omitted).
Two useful references:
O. Gorodetsky, New representations for all sporadic Ap'ery-like sequences, with applications to congruences, arXiv:2102:2102.11839 (2021)
and
Y. Yang, Ap'ery limits and special values of $L$-functions, J. Math. Anal. Appl. {\bf 343} (2008), 492--513.