Relating Gamma and factorial function for non-integer values. We have
$$\Gamma(n+1)=n!,\ \ \ \ \ \Gamma(n+2)=(n+1)!$$
for integers, so if $\Delta$ is some real value with
$$0<\Delta<1,$$
then
$$n!\ <\ \Gamma(n+1+\Delta)\ <\ (n+1)!,$$
because $\Gamma$ is monotone there and so there is another number $f$ with
$$0<f<1,$$
such that
$$\Gamma(n+1+\Delta)=(1-f)\times n!+f\times(n+1)!.$$

How can we make this more precise? Can we find $f(\Delta)$?
Or if we know the value $\Delta$, which will usually be the case, what $f$ will be a good approximation?


Well, $\Gamma(1) = \Gamma(2) = 1$, but $\Gamma(\frac{3}{2}) = \frac{\sqrt{\pi}}{2} <1$, so presumably you need $n>1$.
And $f(n, \Delta) = \frac{\Gamma(n+1+\Delta)-\Gamma(n+1)}{\Gamma(n+2)-\Gamma(n+1)}$.
