"Infinitely Often" Transcendence of Gamma-Function Derivatives

Abstract

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.