Now that some of the previously MSE formulae that I left [here][1] have been applied [Dec.2023][2] to compute high precision record values ($10^{12}$ decimal digits) of trascendental constants $\Gamma(1/3)$ ([Eq.B.IV.8][1]) and $\Gamma(1/4)$ ([Eq.A.IIIa.4 and Eq.A.III.3][1]) as it is reported in ([a][3]) and ([b][4])-([c][5]), 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. 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][6] $$ C_s = - \frac{4d}{\log|\rho|}.$$ 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][7].
**B.** For **$\log(2)$**
$$\begin{equation*}\log(2)=\frac{1}{3}\sum_{n=1}^\infty\left(\frac{1}{3^{3}\cdot2^{13}}\right)^n\,\frac{686430\,n^3 - 742257\,n^2 + 223397\,n - 13858}{n(2n-1)(3n-1)(3n-2)}\,\left[\begin{matrix}
1&\frac{1}{2}&\frac{1}{3}&\frac{2}{3}\\
\frac{1}{12}&\frac{5}{12}&\frac{7}{12}&\frac{11}{12}\\
\end{matrix}\right]_n\tag{2}\label{2}
\end{equation*}$$
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][8].
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][9] as it is reported [here][10]
**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 combinations 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?
[1]: https://math.stackexchange.com/questions/4793437/are-these-fast-convergent-series-for-the-lemniscate-constant-and-gamma-frac1
[2]: http://www.numberworld.org/y-cruncher/#Records
[3]: http://www.numberworld.org/y-cruncher/records/2023_12_14_gamma1d3.txt
[4]: http://www.numberworld.org/y-cruncher/records/2023_12_11_gamma1d4.txt
[5]: http://www.numberworld.org/y-cruncher/records/2023_12_12_gamma1d4.txt
[6]: http://www.numberworld.org/y-cruncher/internals/binary-splitting.html
[7]: https://arxiv.org/pdf/2207.02501.pdf
[8]: http://www.numberworld.org/y-cruncher/internals/formulas.html
[9]: https://flintlib.org/
[10]: https://github.com/flintlib/flint/pull/1767