Playing around with [this series](https://mathoverflow.net/questions/289662/series-involving-gamma-function) for natural values of $a,b$, it appears that more generally for $c\in\mathbb N$, $$\sum\_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+ k+1)! }=\frac{a!\ b!\ (c-1)!}{(a+c)!(b+c)!}$$ and obviously the factorials should be extendable to Gamma functions for all $a,b,c>0$ or even $\Re(a),\Re(b),\Re(c)>0$.  

Moreover when introducing a variable $z\in[-1,1]$, it seems that for rational $z$
$$\sum\_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+k+1)! }z^k=p\log(1-z)+q$$ with rational $p,q$ (depending of course not only on $a,b,c$, but also on $z$).


I don't think this kind of identities is new, as they seem too elementary for that. But I can't find anything related, though Abramowitz & Stegun or Ryzhik should have them. (?)



Series involving factorials
The sum
$$\sum\_{k=0}^\infty \frac{(a+k)!\,(b+k)!}{k!\,(a+b+c+k+1)!}z^k.$$
is not only a generalized hypergeometric series; it's [the original ungeneralized Gauss hypergeometric series](https://en.wikipedia.org/wiki/Hypergeometric_function),
$$\frac{\Gamma(a+1)\,\Gamma(b+1)}{\Gamma(a+b+c+2)}{}\_2F\_1\left({a+1,b+1\atop a+b+c+2}\ \Big |\, z\right).$$
Chapter 15 of [Abramowitz and Stegun](http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf) is on hypergeometric functions. Gauss's theorem (your first formula) is 15.1.20.
For your second formula, see formulas 15.3.11 to 15.3.14 of [Abramowitz and Stegun](http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf).


