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Now that some of the previously MSE formulae that I left [here][3] have been applied [Dec.2023][1] to compute high precision record values ($10^{12}$ decimal digits) of trascendental constants $\Gamma(1/3)$ ([Eq.B.IV.8][3]) and $\Gamma(1/4)$ ([Eq.A.IIIa.4 and Eq.A.III.3][3]) as it is reported in ([a][4]) and ([b][5])-([c][6]), I have undertaken the search for highly efficient series $s$ to calculate $\log(2)$, $\log(3)$ and $\log(5)$ that are computed by the binary splitting method. I have found one conjectured series for $s=\log(3)$, two for $s=\log(2)$ and one for $s=\log(5)$. I am not sure if any of them has been already published, so the question is very simple and the same as the former note: Is any of these series known?. If they are not, I am interested to know a proof for them. Perhaps by means of Wilf-Zeilberger pairs. (See answer below, updated on Nov 13 2024). Any suggestions on this way are welcome.
We use the following notation, where the constant $s$ is expressed as $$s=\sum_{n=1}^\infty\,\rho^n\cdot\frac{p(n)}{r(n)}\cdot\left[\begin{matrix}
a & b & c & ... & z \\
A & B & C & ... & Z \\
\end{matrix}\right]_n=\sum_{n=1}^\infty\frac{p(n)}{r(n)}\cdot\prod_{k=1}^n\frac{r(k)}{q(k)}$$ here $p(n),q(n),r(n)$ are polynomials non vanishing for $n\in\mathbb{N}$, $q(n)$ and $r(n)$ have the same degree $d$ and the convergence ratio $|\rho|$ is the absolute value of the ratio of the leading terms of $r(n)$ and $q(n)$. The ratio of products of Pochhammer's symbols (rising factorials) is written as
$$\left[\begin{matrix}
a & b & c & ... & z \\
A & B & C & ... & Z \\
\end{matrix}\right]_n=\frac{(a)_n(b)_n(c)_n ... (z)_n}{(A)_n(B)_n(C)_n ... (Z)_n} $$ where the degree $d$ is the number of elements in a row (they are the same for both rows) and $$(w)_n = \frac{\Gamma(w+n)}{\Gamma(w)}=w(w+1)(w+2)...(w+n-1)$$
The computational speed is measured through the [binary splitting cost][7] $$ C_s = - \frac{4d}{\log|\rho|}.$$ This allows to (asymptotically) rank, classify and compare different hypergeometric-type algorithms by performance.
I came across these expressions. Three of them look pretty simple Ramanujan type formulae,
**A.** For **$\log(3)$**
$$\begin{equation*}\log(3)=\sum_{n=1}^\infty\left(\frac{1}{3^{5}}\right)^n\cdot\frac{88\,n-14}{n(2n-1)}\cdot\left[\begin{matrix}
1&\frac{1}{2}\\
\frac{1}{6}&\frac{5}{6}\\
\end{matrix}\right]_n\tag{1}\label{1}
\end{equation*}$$ It has a binary splitting cost $C_s=\frac{8}{5\,\log(3)}=1.4638..$. In preliminary tests this expression performs faster than the fastest known series for such constant that is based on a 4-term Machin-like arcotanh formula with arguments 251, 449, 4801 and 8749. See Table 1 [here][8].
**B.** For **$\log(2)$**
$$\begin{equation*}\small{\log(2)=\sum_{n=1}^\infty\left(\frac{1}{3^{3}\cdot2^{13}}\right)^n\,\frac{686430\,n^3 - 742257\,n^2 + 223397\,n - 13858}{3n(2n-1)(3n-1)(3n-2)}\,\left[\begin{matrix}
1&\frac{1}{2}&\frac{1}{3}&\frac{2}{3}\\
\frac{1}{12}&\frac{5}{12}&\frac{7}{12}&\frac{11}{12}\\
\end{matrix}\right]_n}\tag{2}\label{2}
\end{equation*}$$
It has a binary splitting cost $C_s=\frac{16}{\log(3^{3}\cdot2^{13})}=1.3001..$. Preliminary tests show that this series performs slightly faster than the fastest known series for such constant that is based on a 3-term Machin-like arcotanh formula with arguments 26, 4801 and 8749. See [here][9].
The third one is,
$$\begin{equation*}\log(2)=\frac{1}{2}\,\sum_{n=1}^\infty\left(\frac{1}{3^{5}\cdot2^{4}}\right)^n\cdot\frac{1794\,n-297}{n(2n-1)}\cdot\left[\begin{matrix}
1&\frac{1}{2}\\
\frac{1}{6}&\frac{5}{6}\\
\end{matrix}\right]_n\tag{3}\label{3}
\end{equation*}$$ This expression has a binary splitting cost $C_s=\frac{8}{\log(3^{5}\cdot2^{4})}=0.96786..$. Preliminary tests show that this series performs much faster than the fastest known series for such constant that is based on the mentioned 3-term Machin-like formula.
$\log(2)$ is an important fundamental constant and this last expression, being very efficient, should be taken as a standard high precision formula for this constant to be included inside mathematical software whenever it is implemented as a binary splitting algorithm. In fact this formula will be part of [FLINT][10] as it is reported [here][11]
**C.** For **$\log(5)$**
$$\begin{equation*}\log(5)=\sum_{n=1}^\infty\left(\frac{-1}{3^{3}\cdot5^2}\right)^n\cdot\frac{364\,n-62}{-n(2n-1)}\cdot\left[\begin{matrix}
1&\frac{1}{2}\\
\frac{1}{6}&\frac{5}{6}\\
\end{matrix}\right]_n\tag{4}\label{4}
\end{equation*}$$ It has a binary splitting cost $C_s=\frac{8}{\log(675)}=1.2280..$. Preliminary tests show that this series performs pretty faster than the fastest known series for such constant that is based on a 4-term Machin-like formula (a linear combination of arcotanhs with arguments 251, 449, 4801 and 8749).
Eqs.(3-4) allow also to get high precision values of $\log(10)$ that is an important constant in numerical analysis.
Q: Is any of Eqs.(1-4) known? and if they are not, would it be possible to get the proofs?
I would like to thank Dr. J. Guillera for encouraging me to investigate finding these formulas. To H. Cohen and the PARI GP team at the University of Bordeaux for this excellent parallelizable search tool and to Jordan Ranous at Storage Review for providing me with high-performance multicore facilities.
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**Updated on Feb.16.2024**
Fortunately for three of these series, Eqs. (1), (3) and (4), I have found a proof using classical methods which I placed in the Answers section below. (I doubt that there is a proof based on modular forms, this transformation of coefficients for the logarithm is not in PSL2(ℝ).)
Eq.(3) (now proven) was used on Feb.12 to double up to $3$ x $10^{12}$ the known number of decimal digits places of log(2), Eq.(2) (unproven) was used for verifying, as it is reported [here][1]. Details are found in this [log][2]. Formulas below are unpublished and are just placed here at MO.
**Updated on Jul.19.2024**
At the end of my response in the answer's section below, I have placed a new proof based on some closed forms found in a recently published work (2024) by by Zhi-Wei Sun and Yajun Zhou arXiv:2401.14197v1 [math.CA], 25 Jan 2024.
**Updated on Nov.13.2024**
**D.** New series for **$\log(2)$**
The following are two additional series for $\log(2)$ that are even more efficient than Eq.(2) under the application of the binary splitting algorithm,
$$\begin{equation*}\small{\log(2)=\sum_{n=1}^\infty\left(\frac{1}{2^4\cdot3^2\cdot5^5}\right)^n\,\frac{3927264\,n^3 - 4300512\,n^2 + 1209726\,n - 81891}{4n(2n-1)(4n-1)(4n-3)}\,\left[\begin{matrix}
1&\frac{1}{2}&\frac{1}{4}&\frac{3}{4}\\
\frac{1}{10}&\frac{3}{10}&\frac{7}{10}&\frac{9}{10}\\
\end{matrix}\right]_n}\tag{5}\label{5}
\end{equation*}$$ It has a binary splitting cost $Cs=1.2291..$ performing very fast. (This series was privately communicated by MO user @xiaoshuchong).
The next series was found using the PSLQ algorithm. It has a cost $Cs=1.1335..$ providing the currently 2nd fastest formula to compute this constant,
$$\begin{equation*}\small{\log(2)=\sum_{n=1}^\infty\left(\frac{1}{2^4\cdot3^3\cdot5^5}\right)^n\,\frac{13885704\,n^3 - 15397068\,n^2 + 4353342\,n - 295245}{2n(2n-1)(6n-1)(6n-5)}\,\left[\begin{matrix}
1&\frac{1}{2}&\frac{1}{6}&\frac{5}{6}\\
\frac{1}{10}&\frac{3}{10}&\frac{7}{10}&\frac{9}{10}\\
\end{matrix}\right]_n}\tag{6}\label{6}
\end{equation*}$$
It is interesting that $\log(2)$ series Eqs.(3), (5) and (6) can be proven by means of Wilf-Zeilberger $(F,G)$ pairs starting from a proper WZ seed and companion $F(n,k)$. The proof will be located in the answer below.
[1]: http://www.numberworld.org/y-cruncher/#Records
[2]: http://www.numberworld.org/y-cruncher/records/2024_1_27_log2.txt
[3]: https://math.stackexchange.com/questions/4793437/are-these-fast-convergent-series-for-the-lemniscate-constant-and-gamma-frac1
[4]: http://www.numberworld.org/y-cruncher/records/2023_12_14_gamma1d3.txt
[5]: http://www.numberworld.org/y-cruncher/records/2023_12_11_gamma1d4.txt
[6]: http://www.numberworld.org/y-cruncher/records/2023_12_12_gamma1d4.txt
[7]: http://www.numberworld.org/y-cruncher/internals/binary-splitting.html
[8]: https://arxiv.org/pdf/2207.02501.pdf
[9]: http://www.numberworld.org/y-cruncher/internals/formulas.html
[10]: https://flintlib.org/
[11]: https://github.com/flintlib/flint/pull/1767