**I-a. Some functions**
As we will use these in the continued fraction evaluations below, recall the *Riemann zeta function* $\zeta(s),$ and *[Dirichlet beta function][1]* $\beta(s),$
\begin{align}
\zeta(s) &= \sum_{n=1}^\infty\frac{1}{n^s}\\
\beta(s) &= \sum_{n=1}^\infty\frac{(-1)^{n-1}}{(2n-1)^s}
\end{align}
and special cases of the *[Clausen function][2]* $\operatorname{Cl}_s(x),$
$$\operatorname{Cl}_2(x) = \sum_{n=1}^\infty\frac{\sin(n\,x)}{n^2}$$
\begin{align}
K &=\operatorname{Cl}_2\left(\tfrac12\pi\right) = \beta(2) = \sum_{n=0}^\infty\frac{1}{(4n+1)^2}-\sum_{n=0}^\infty \frac{1}{(4n+3)^2} \\
\kappa &= \operatorname{Cl}_2\left(\tfrac13\pi\right) \,=\, \frac{3\sqrt{3}}{4} \left(\sum_{n=0}^\infty\frac{1}{(3n+1)^2}-\sum_{n=0}^\infty \frac{1}{(3n+2)^2} \right)
\end{align}
with *Catalan's constant* $K$ and its cubic counterpart *[Gieseking's constant][3]* $\kappa$.
---
**I-b. 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 in $n$ of form,
$$(n+1)^2u_{n+1} = (\color{blue}{an^2+an+b})u_k+\color{blue}c\,n^2u_{n-1}$$
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.
---
**II. Degree 2**
Define the polynomial function,
$$s_n = \color{blue}{an^2+an+b}$$
and the continued fraction,
$$C_2(a,b,c)=\cfrac{1}{s_0 + \cfrac{1^4\, \color{blue}c}{s_1 + \cfrac{2^4\, \color{blue}c}{s_2+ \cfrac{3^4\,\color{blue}c}{s_3+\ddots } }}}$$
**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. The first evaluation is valid since it was found by Apery, while the second (in green) is courtesy of H. Cohen's answer below (though it has *slow* convergence which is why I missed it).
---
**III. Degree 3**
In [Cooper's paper][4], we find the recurrence relation with deg-$3$ coefficients in $n$,
$$(n+1)^3 v_{n+1} = \color{blue}{-(2n+1)(an^2+an+a-2b)}v_n \color{blue}{- (a^2+4c)}n^3v_{n-1}$$
where Zagier's $(a,b,c)$ also apply. Define the polynomial function,
$$t_n = \color{blue}{-(2n+1)(an^2+an+a-2b)}$$
and the continued fraction with constant $\color{blue}{d = -(a^2+4c)}$,
$$C_3(a,b,c)=\cfrac{1}{t_0 + \cfrac{1^6\, \color{blue}d}{t_1 + \cfrac{2^6\, \color{blue}d}{t_2+ \cfrac{3^6\,\color{blue}d}{t_3+\ddots } }}}$$
**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 $-d=125, 1, 64, 81, 16, -27,$ 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 $\zeta(3)$.
---
**IV. Degree 4**
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?*
---
**V. Degree 5**
But Zudilin found a 3-term recurrence,
$$Q_1(n) v_{n+1} = Q_2(n) v_n + Q_3(n) v_{n-1}$$
where $Q_i$ are polynomials of deg-$5$ and used it in an analogous continued fraction for $\zeta(4)$,
$$C_4(a_1, a_2,\dots a_n) =\cfrac{1}{p_0 + \cfrac{1^8\, q_1}{p_1 + \cfrac{2^8\, q_2}{p_2+ \cfrac{3^8\,q_3}{p_3+\ddots } }}}$$
and where $p_i, q_i$ are polynomial functions. (*To be discussed in the next post.*)
---
**VI. Questions**
1. Are all cfracs with proposed closed-forms correct? (I know two of them are.)
2. What are the closed-forms of the others?
[1]: https://mathworld.wolfram.com/DirichletBetaFunction.html
[2]: https://mathworld.wolfram.com/ClausenFunction.html
[3]: https://mathworld.wolfram.com/GiesekingsConstant.html
[4]: https://www.researchgate.net/publication/257642843_Sporadic_sequences_modular_forms_and_new_series_for_1p