"Infinitely Often" Transcendence of Gamma-Function Derivatives

Abstract

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