Playing around with this series for natural values of $a,b$, it appears that more generally for $c\in\mathbb N$, $$\sum_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+1)! }=\frac{a!\ b!\ (c-1)!}{(a+c)!(b+c)!}$$ 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 $z$ $$\sum_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+1)! }z^k=p\log(1-z)+q$$ 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. (?)

Playing around with this series 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. (?)