I have an optimization problem that involves an Stirling number term $S(n,k;a,b,0)$. The partial derivatives of the "$\log S(n,k;a,b,0)$" will be with respect to all four non-zero parameters $\theta=\{n,k,a,b\}$:
$$\frac{\partial\log S(n,k;a,b,0)}{\partial\theta} = \frac{\frac{\partial S(n,k;a,b,0)}{\partial\theta}}{S(n,k;a,b,0)}$$
as-is, this is a pretty nasty derivative to deal with. I wonder if the derivatives of the Stirling numbers have been studied any where? I am considering both forms of Stirling number representation: recursive, or explicit.
[I am dreaming of a nice and tight bound around the Stirling number and dealing with taking the derivatives over that bound.]