# On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"

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 $$\kappa$$.

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

1. Are all cfracs with proposed closed-forms correct? (I know two of them are.)
2. What are the closed-forms of the others?
• Gieseking's constant $\kappa$ seems not so well-known compared to its cousin Catalan's constant $K$. But both are also volumes, the former for the hyperbolic Gieseking 3-manifold, while the latter is 1/4 for the hyperbolic octahedron. Fortunately, I made a large list of contexts for $\kappa$ and $K$ back in 2019 which is why I suspected both might appear in the deg-$2$ continued fractions. Kindly see this MSE post. May 20 at 6:48
• Is there any reason or general pattern for the sign of the continued fraction ? For example it seems to be always + for $\zeta(even)$ and - for $\zeta(odd)$ ? May 20 at 11:46
• @CHUAKS Not always. In the list, there are 3 cfracs for $\zeta(3)$. The original one by Apery is positive. May 20 at 11:55

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.

• I just re-tested $C_2(-17,-6,-72)$. I figured out the answer: it converges VERY slowly compared to the others. May 19 at 15:46
• I just read Gorodetsky's 2021 paper. He mentions 6+6+3 = 15 sequences, but seems to have missed a 16th one found by Zudilin way back in 2002. Kindly see new MO post. May 20 at 10:09

Zagier answered your first question in his original paper. See the table in p. 11 here. He gives there evaluations of the continued fractions associated with his sequences A, C, D, E and F (note that you use the ordering D, F, C, A, E) which agree with your findings. He does not evaluate the one associated to B (and I am not aware of it being evaluated elsewhere).

• Thank you for the reference to Zagier's paper. It is good to know I got the evaluations right. You're the "O. Gorodetsky" in the post after this, I assume? By the way, I believe there are $15+1=16$ known sporadic sequences, if we include Zudilin's recurrence for $\zeta(4)$ (call it $Z$) which I discussed in that post. However, there is no Ramanujan-Sato pi formula associated with $Z$ unlike the other 15. Cooper found $t_7, t_{10}, t_{18}$, but not yet $t_{30}$ so there might still be another "sporadic sequence" out there. May 21 at 15:17
• @TitoPiezasIII I agree that there are some other sequences that may be dubbed 'sporadic', depending on your interpretation of 'sporadic' (I'll add that these specific 15 sequences were put under the same 'sporadic' umbrella by various authors; my paper did not introduce any new convention). However, each of the 15 sequences was found in a (wide but finite) search within an infinite family of recurrences, making the adjective 'sporadic' well deserved in my view. My understanding is that the 16th sequence you mention was not found by searching within a well-defined family of recurrences. May 21 at 16:20
• I should also say that all the 15 sequences were found to have modular origins; I do not know if the 16th sequence can be explained in this way. May 21 at 16:26
• I tried to evaluate the cfrac associated with Zagier's sequence $B$. It seems to have peculiar properties such as having 6 limits, one of which has a closed-form, though I am not sure. Kindly see MO post. May 22 at 17:56
• I found four sequences with modular origins apparently of the same cubic type found by Almkvist-Zudilin. Kindly see this post. May 28 at 17:42