Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing? As per https://dlmf.nist.gov/8.10#E13 we have
$$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma%
\left(n,n-1\right)}{\Gamma\left(n\right)}$$ for $n=1,2, \ldots$.
My question is: replacing $n$ by real $s>1$, are these monotone in $s$?
Note: For large $n$ both of these converge to $\frac{1}{2}$: rewriting via repeated integration by parts we get $\frac{\Gamma%
\left(n,n-1\right)}{\Gamma\left(n\right)}=\exp\{-(n-1)\}\sum_{i=0}^{n-1} \frac{(n-1)^i}{i!}$ which is the CDF of Poisson distribution with $\lambda=n-1$ evaluated at $x=\lambda=n-1$; as $n$ increases this approaches CDF of normal with mean and variance $\lambda$, which at $x=\lambda$ is $\frac{1}{2}$. (I would be interested in alternative proofs of this fact as well.)
 A: Concerning the monoticity, I have serious problem to prove it (even it seems to be obvious) since the first derivatives involve the Meijer G function).
For the remaining, at least for "large" values of $x$, we can write that
$$\Gamma (x,x-\epsilon )= \Gamma (x,x)+e^{-x} x^{x-1} \epsilon +\frac{1}{2} e^{-x} x^{x-2} \epsilon
   ^2-\frac{1}{6} \left(e^{-x} (x-2) x^{x-3}\right) \epsilon ^3+O\left(\epsilon ^4\right)$$
Making $\epsilon=1$
$$\Gamma (x,x-1)\sim\Gamma (x,x)+\frac{1}{3} e^{-x} \left(3 x^2+x+1\right) x^{x-3}$$ On the other hand, as given by $DLMF$, for large $x$
$$\Gamma (x,x)= x^{x-1}e^{-x} \left(\sqrt{\frac{\pi }{2}} \sqrt{x}-\frac{1}{3}+\frac{\sqrt{\frac{\pi }{2}}}{12
   \sqrt{x}}-\frac{4}{135 x}+\cdots\right)$$
Using the simplest form of Stirling approximation, we then have
$$\frac{\Gamma (x,x)}{\Gamma (x)}=\frac{1}{2}-\frac{t}{3 \sqrt{2 \pi }}-\frac{t^3}{540 \sqrt{2 \pi }}-\frac{t^4}{576}+O\left(t^{5}\right)$$
$$\frac{\Gamma (x,x-1)}{\Gamma (x)}=\frac{1}{2}+\frac{1}{3} \sqrt{\frac{2}{\pi }} t+\frac{67 t^3}{270 \sqrt{2 \pi
   }}-\frac{t^4}{576}++O\left(t^5\right)$$ where $t=\frac 1 {\sqrt x}$.
Using these truncated expansions for $x=9$, they give respectively $$\frac{18894947-2100714 \sqrt{\frac{2}{\pi }}}{37791360}\approx 0.455628 \qquad \text{while} \qquad \frac{\Gamma (9,9)}{\Gamma (9)}\approx 0.455653$$
$$\frac{23327}{46656}+\frac{1687}{7290 \sqrt{2 \pi }}\approx 0.592299\qquad \text{while} \qquad \frac{\Gamma (9,8)}{\Gamma (9)}\approx 0.592547$$
