**I-a. Some functions** As these will be used 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} \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][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, $$(n+1)^2\,u_{n+1} = (an^2+an+b)u_k+ cn^2\,u_{n-1}$$ 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. --- **I-c. 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 $F_k(n)$ are polynomials ***all*** of degree $m$. Define two polynomial functions using the rules, \begin{align} p(n) &= F_2(n)\\ q(n) &= F_1(n)\, F_3(n+1) \end{align} which implies $q(n)$ has degree ***twice*** that of $p(n)$. Define the continued fraction, $$C=\cfrac{1}{p(0) + \cfrac{q(0)}{p(1) + \cfrac{q(1)}{p(2)+ \cfrac{q(2)}{p(3)+\ddots } }}}$$ then it seems $C$ may have a nice closed-form based on the properties of the recurrence relation. Examples below. --- **II. 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}{an^2+an+b}\\ q(n) &= \color{blue}{(n+1)^2}\times \color{blue}{c(n+1)^2} = c(n+1)^4 \end{align} Then define the cfrac, $$C_2(a,b,c)=\cfrac{1}{p(0) + \cfrac{q(0)}{p(1) + \cfrac{q(1)}{p(2)+ \cfrac{q(2)}{p(3)+\ddots } }}}$$ **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. *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 (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 $n$, $$(n+1)^3\,v_{n+1} = -(2n+1)(an^2+an+a-2b)v_n +(-a^2-4c)n^3\,v_{n-1}$$ and Zagier's $(a,b,c).$ Using the same rules, let, \begin{align} r(n) &= -(2n+1)(an^2+an+a-2b)\\ s(n) &= (n+1)^3\times(-a^2-4c)(n+1)^3 = -(a^2+4c)(n+1)^6 \end{align} Define the cfrac, $$C_3(a,b,c)=\cfrac{1}{r(0) + \cfrac{s(0)}{r(1) + \cfrac{s(1)}{r(2)+ \cfrac{s(2)}{r(3)+\ddots } }}}$$ **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 $d = a^2+4c =125, 1, 64, 81, 16, -27,$ 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 $\zeta(3)$. --- **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 $\zeta(4).$ (*To be discussed in the [next post][5].*) --- **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 [5]: https://mathoverflow.net/q/447166/12905