Relatively little is known about the arithmetic properties of Gamma-function derivatives evaluated at arbitrary points $q\in\mathbb{ Q}\setminus\mathbb{ Z}_{ \leq0}$. In recent work, we showed that the sequence $\left\{ \Gamma^{ \left(n\right)}\left(1\right)\right\}_{ n\geq1}$ contains transcendental elements infinitely often. That result is now generalized to all sequences $\left\{ \Gamma^{ \left(n\right)}\left(q\right)\right\}_{ n\geq1}$ for $q\in\tfrac{ 1}{ 2}\mathbb{ Z}\setminus\mathbb{ Z}_{ \leq0}$. Moreover, for all such $q$ we derive a lower bound, $\beta\left(N\right)=\max\left\{ 0,\sqrt{ N}-5/2\right\}/N$, for the density of transcendental elements $\Gamma^{ \left(n\right)}\left(q\right)$ among $n\in\left\{ 1,2,\ldots,N\right\}$, where $\beta\left(N\right)\asymp N^{ -1/2}\rightarrow0$ as $N\rightarrow\infty$. For $q\in\mathbb{ Q}\setminus\tfrac{ 1}{ 2}\mathbb{ Z}$, we find the somewhat weaker result that at least one of the sequences $\left\{ \Gamma^{ \left(n\right)}\left(q\right)\right\}_{ n\geq1}$, $\left\{ \Gamma^{ \left(n\right)}\left(1-q\right)\right\}_{ n\geq1}$ contains infinitely many transcendental elements.
Categories: math.NT