"Infinitely Often" Transcendence of Gamma-Function Derivatives

TL;DR

This paper proves that for certain rational points, the derivatives of the Gamma function are transcendental infinitely often, and provides bounds on their density among natural numbers.

Contribution

It generalizes previous results to all sequences of Gamma derivatives at specific rational points and establishes lower bounds on the density of transcendental values.

Findings

Sequences of Gamma derivatives at certain rational points contain infinitely many transcendental elements.

A lower bound on the density of transcendental derivatives among the first N derivatives is established.

For other rational points, at least one of two related sequences contains infinitely many transcendental 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.