We have Ramanujan's well-known,
$$\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots\color{blue}+\,\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\ddots}}}}$$
However, as discussed in this post, the series can also be expressed as a nice continued fraction. And since $\Gamma\big(\tfrac12\big) = \sqrt{\pi}$, we can then express the identity as,
$$\Gamma\big(\tfrac12\big)\sqrt{\frac{e}{2}} = 1+\cfrac{2/2}{2+\cfrac{3/2}{3+\cfrac{4/2}{4+\cfrac{5/2}{5+\ddots}}}} \; \color{blue}+ \; \cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\ddots}}}}$$
It turns out it may have a cubic counterpart,
$$\Gamma\big(\tfrac13\big)\sqrt[3]{\frac{e}{9}} = 1+\cfrac{2/3}{2+\cfrac{3/3}{3+\cfrac{4/3}{4+\cfrac{5/3}{5+\ddots}}}} \; \color{blue}+ \; \cfrac1{1+\cfrac{2}{1+\cfrac{3}{1+\cfrac{\color{red}5}{1+\ddots}}}}$$
where the 4th continued fraction is missing numerators $P(n)=3n+1 = 4,7,10,13,\dots$ The four cfracs apparently have closed-forms as,
\begin{align} \Gamma\big(\tfrac12\big)\sqrt{\frac{e}{2}} &= \sqrt{\frac{e}{2}}\times\Big(\Gamma\big(\tfrac12\big)-\Gamma\big(\tfrac12,\tfrac12\big)\Big)\; \color{blue}+ \, \sqrt{\frac{e}{2}}\times\Big(\Gamma\big(\tfrac12,\tfrac12\big)\Big)\\[6pt] \Gamma\big(\tfrac13\big)\sqrt[3]{\frac{e}{9}} &= \sqrt[3]{\frac{e}{9}}\times\Big(\Gamma\big(\tfrac13\big)-\Gamma\big(\tfrac13,\tfrac13\big)\Big)\; \color{blue}+ \; \sqrt[3]{\frac{e}{9}}\times\Big(\Gamma\big(\tfrac13,\tfrac13\big)\Big) \end{align}
and their decimal expansions are A060196, A108088, A108744, A108745, respectively.
Questions:
- Given Pochhammer symbol $(x)_n$, how do we prove that,
\begin{align} \sum_{n=1}^\infty \frac{1}{2^n (\frac12)_n} &= \sqrt{\frac{e}{2}} \times\,\Gamma\big(\tfrac12\big)\,\operatorname{erf}\Big(\sqrt{\tfrac 12}\Big)\\ &= \sqrt{\frac{e}{2}}\times\Big(\Gamma\big(\tfrac12\big)-\Gamma\big(\tfrac12,\tfrac12\big)\Big) = 1+\cfrac{2/2}{2+\cfrac{3/2}{3+\cfrac{4/2}{4+\cfrac{5/2}{5+\ddots}}}} \end{align}
- Similarly, how do we show that,
$$\sum_{n=1}^\infty \frac{1}{3^n (\frac13)_n} = \sqrt[3]{\frac{e}{9}}\times\Big(\Gamma\big(\tfrac13\big)-\Gamma\big(\tfrac13,\tfrac13\big)\Big) = 1+\cfrac{2/3}{2+\cfrac{3/3}{3+\cfrac{4/3}{4+\cfrac{5/3}{5+\ddots}}}}$$
P.S. Part of Question 1 has already been answered in this post, but I included it for comparison and to see if an alternative or modified proof can be found that covers both Question 1 and 2.