I. Some functions
As we will use these in the continued fraction evaluations below, recall the Riemann zeta function $\zeta(s),$ Dirichlet beta function $\beta(s),$ \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 $\operatorname{Cl}_n(x),$
\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 $\kappa$.
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 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.
III. 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{red}{??}\\ 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)=\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.
IV. Degree 3
In Cooper's paper, 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 $\zeta(3)$.
V. Degree 4
Zudilin also found an analogous continued fraction for $\zeta(4)$,
$$C_4(a_1, a_2,\dots a_n) =\cfrac{1}{p_0 + \cfrac{1^8\, q_1}{p_1 + \cfrac{2^8\, q_2}{p_2+ \cfrac{3^8\,q_3}{p_3+\ddots } }}}$$
where $p_i, q_i$ are polynomial functions. Makes you wonder if there is for $\beta(4) = \frac1{768}\left[\psi_3(\frac14)-8\pi^4\right]$ as well.
VI. Questions
- Are all cfracs with proposed closed-forms correct? (I know two of them are.)
- What are the closed-forms of the others?