**UPDATE Feb.02.2024**

The series below, Eq.(3) for computing and Eq.(2) for verifying, were applied by Andrew Sun on Dec.22.2023 to get over $2\cdot10^{12}$ decimal digits and break the number of currently known decimals of Apery's Constant. The whole process took about 12 days as it is reported in [1], [2] and [3].

It is worth to point out that these proven fast ζ(3) formulae Eqs.(1,2 & 3) are un-published and they have been only placed here in MO.

By means of a generalized search of hypergeometric-type series based on Dougall's sum and applying the Wilf-Zeilberger method as implemented by Jesús Guillera to prove some classical constants relationships, three proven series are found for Apéry's constant $\zeta(3)$. These series show the lowest known computing costs to apply the binary splitting method using A. Yee's y-cruncher software, a high performance platform to compute a huge number of digits of some classical constants, having set several records in this regard. These series are currently the fastest known to calculate an arbitrary number of digits for this constant.

I am not sure if any of these series has been published, so the question is simple.

Are these series known?

The first one is $$\begin{equation*}\zeta(3) = \frac{1}{12}\sum_{n=1}^{\infty}\frac{P(n)}{n^5(2n-1)(3n-1)(3n-2)(4n-1)(4n-3)\binom{3n}{n}^2\binom{6n}{3n}^2\binom{7n}{3n}}\tag{1}\label{1} \end{equation*}$$ with $$P(n) = 376698240\,n^7 - 935880048\,n^6 + 938536090\,n^5 - 491213175\,n^4 + 144859115\,n^3 - 24133842\,n^2 + 2114280\,n - 75600$$ It has convergence rate $\rho = 1/22235661$ giving about $7.35$ decimal digits per term. The computing cost $C_s$ is measured through $$C_s = - \frac{4\,d}{\log\rho}$$ where $d$ is the denominator's polynomial degree. In this case $d = 10$. This gives $C_s=2.3645...$.

The second series has $C_s = 2.3070...$ with $d=14$ and $\rho=1/34828517376$ giving about $10.54$ decimal digits per term. It is $$\begin{equation*}\zeta(3) = \frac{1}{24}\sum_{n=1}^{\infty}\frac{P(n)}{R(n)\cdot\binom{3n}{n}\binom{6n}{3n}\binom{8n}{4n}\binom{9n}{3n}\binom{10n}{5n}}\tag{2}\label{2} \end{equation*}$$ where $$P(n)= 250765325100000\,n^{11} - 1087318449630000\,n^{10} + 2067749814046250\,n^9 - 2269551612681475\,n^8 + 1592180015776565\,n^7 - 746938801646725\,n^6 + 238210943593421\,n^5 - 51452348050672\,n^4 + 7352050259484\,n^3 - 660416507568\,n^2 + 33552610560\,n - 731566080$$ and $$R(n)=n^5\,(2n-1)\,(3n-1)\,(3n-2)\,(4n-1)\,(4n-3)\,(5n-1)\,(5n-2)\,(5n-3)\,(5n-4)$$

The 3rd series has $C_s=2.0514...$ with $d=14$ and $\rho=-1/717445350000$ giving about $11.86$ decimal digits per term. It can be expressed as $$\begin{equation*}\zeta(3) = \frac{1}{48}\sum_{n=1}^{\infty}\frac{(-1)^{(n-1)}\,P(n)}{R(n)\cdot\binom{5n}{n}\binom{5n}{2n}\binom{9n}{4n}\binom{10n}{5n}\binom{12n}{6n}}\tag{3}\label{3} \end{equation*}$$ where $$P(n)= 1565994397644288\,n^{11} - 6719460725627136\,n^{10} + 12632254526031264\,n^9 - 13684352515879536\,n^8 + 9451223531851808\,n^7 - 4348596587040104\,n^6 + 1352700034136826\,n^5 - 282805786014979\,n^4 + 38721705264979\,n^3 - 3292502315430\,n^2 + 156286859400\,n - 3143448000$$ and $$R(n)=n^5\,(2n-1)^3\,(3n-1)\,(3n-2)\,(4n-1)\,(4n-3)\,(6n-1)\,(6n-5)$$

A non-native implementation of such algorithms has been done for y-cruncher software preparing custom input files for each formula. I tested these series under this special platform against some classical formulas that has been used to break the $\zeta(3)$ digits number record. In particular the Amdeberhan-Zeilberger series **AZ** (1997) $C_s=2.8854...$ and Wedeniwski's series **Wed** (1998) $C_s=2.7554...$.

Testings were done on an Intel Core Kaby Lake i5-7300U 3.5 Ghz 32 Gb RAM standard laptop with Windows 11 Pro 64bit OS. The following Table shows timings in seconds to get 50.000.000 decimal digits of Apéry constant. $Arch$ refers to the word size of the formula's input integer polynomial coefficients. $N$ indicates if the series summand has $N$ terms grouped from the basic formula. Timings are the average of 5 runs.

\begin{align*} \begin{array}{|c|c|c|c|c|c|c|} \hline {Series} & {Time\,s} & {Arch.} & N & \frac{dec.\ digits}{term} & C_s & d \\ \hline \mathrm{Eq.(3)} & 23.361 & 64-bit & 1 & 11.86 & 2.0514 & 14\\ \hline \mathrm{Eq.(2)} & 26.387 & 64-bit & 1 & 10.54 & 2.3070 & 14\\ \hline \mathrm{Eq.(1)} & 27.465 & 32-bit & 1 & 7.35 & 2.3645 & 10 \\ \hline \mathrm{Wed-II} & 33.095 & 64-bit & 2 & 10.08 & 2.7554 & 16\\ \hline \mathrm{AZ-III} & 34.697 & 64-bit & 3 & 9.03 & 2.8854 & 15\\ \hline \mathrm{Wed-I} & 34.893 & 32-bit & 1 & 5.04 & 2.7554 & 8\\ \hline \mathrm{AZ-I} & 37.603 & 32-bit & 1 & 3.01 & 2.8854 & 5\\ \hline \end{array} \end{align*}

Considering this limited scope of testings, it is observed that, under the y-cruncher's generic algorithm custom implementation, these -I guess- new series provide a significative increment of computing speed for Apéry's $\zeta(3)$. Eqs.$(1), (2)$ and $(3)$ are about 21%, 24% and 33% faster than classical Wedeniwski's formula respectively.

Finally, the proof of these series is obtained by means of WZ certificates (too long to put them here) through the code in Guillera with input patterns $$entry = 2+7\,n,1+3\,n,1+2\,n,1+1\,n,1+1\,n$$ $$entry = 2+10\,n,1+4\,n,1+3\,n,1+2\,n,1+1\,n$$ $$entry = 2+11\,n,1+4\,n,1+3\,n,1+2\,n,1+1\,n$$ for Eqs.$(1), (2)$ and $(3)$ respectively.