I'm looking for a closed-form expression for $$ \left.\left[\frac{\partial^n}{\partial \nu^n}K_{\nu}(z)\right]\right|_{\nu=\pm\tfrac{1}{2}},\;\;n\ge1 $$ where $K_{\nu}(z)$ denotes the MacDonald function. In case it matters, I'm interested only in the case when $z>0$. For $n=1$, the answer is as follows: $$ \left.\frac{\partial\mathop{K_{\nu}}\nolimits\!\left(x\right)}{\partial\nu}% \right|_{\nu=\pm\frac{1}{2}}=\pm\sqrt{\frac{\pi}{2x}}\mathop{E_{1}}\nolimits% \!\left(2x\right)e^{x}, $$ cf. http://dlmf.nist.gov/10.38.E7. Here $E_1$ denotes the exponential integral.
What about higher-order derivatives $(n\ge2)$?
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