Lower Bounds for Densities of Transcendental Gamma-Function Derivatives

TL;DR

This paper establishes lower bounds on the density of transcendental derivatives of the Gamma function at various lattice and shifted lattice points, extending previous results on their transcendence properties.

Contribution

It provides new bounds on the number of algebraic derivatives of the Gamma function at lattice points and constructs density bounds for their transcendental derivatives.

Findings

At most n-1 algebraic derivatives at positive lattice points for n≥2.

At most n algebraic derivatives at shifted lattice points for n≥1.

Derived lower bounds for densities of transcendental derivatives among specified index sets.

Abstract

In recent work, we showed that for all $q\in\tfrac{1}{2}\mathbb{Z}\setminus\mathbb{Z}_{\leq0}$ the sequence $\left\{\Gamma^{\left(n\right)}\left(q\right)\right\} _{n\geq1}$ contains transcendental elements infinitely often, with the density of transcendental $\Gamma^{\left(n\right)}\left(q\right)$ among $n\in\left\{1,2,\ldots,N\right\}$ bounded below by $\beta\left(N\right)=\max\left\{0,\sqrt{N}-5/2\right\}/N$. For both fixed and variable $n$, we now study the transcendence of $\Gamma^{\left(n\right)}\left(q\right)$ at both positive lattice points $q=m\in\left\{1,2,\ldots\right\}$ and rationally shifted lattice points $q=\widetilde{m}\in\left\{\kappa,\pm1+\kappa,\pm2+\kappa,\ldots\right\}$ (for $\kappa\in\left(0,1\right)\cap\mathbb{Q}$ such that $\Gamma\left(\kappa\right)$ is transcendental). For $n\in\mathbb{Z}_{\geq2}$, we find there are at most $n-1$ algebraic $\Gamma^{\left(n\right)}\left(m\right)$, and for $n\in\mathbb{Z}_{\geq1}$, there are at most $n$ algebraic $\Gamma^{\left(n\right)}\left(\widetilde{m}\right)$ for each one-sided shifted lattice (i.e., $\widetilde{m}\geq\kappa$ or $\widetilde{m}\leq\kappa$). These results form the basis for constructing lower bounds for the densities of transcendental $\Gamma^{\left(n\right)}\left(m\right)$ among $m\in\left\{1,2,\ldots,M\right\}$ and transcendental $\Gamma^{\left(n\right)}\left(\widetilde{m}\right)$ among either $\widetilde{m}\in\left\{\kappa,1+\kappa,2+\kappa,\ldots,M+\kappa\right\}$ or $\widetilde{m}\in\left\{\kappa,-1+\kappa,-2+\kappa,\ldots,-M+\kappa\right\}$. Allowing $n$ to vary, we derive lower bounds for the bivariate densities of both transcendental $\Gamma^{\left(n\right)}\left(m\right)$ among $\left(n,m\right)\in\left\{2,3,\ldots,N\right\}\times\left\{1,2,\ldots,M\right\}$ and transcendental $\Gamma^{\left(n\right)}\left(\widetilde{m}\right)$ among one-sided shifted-lattice analogues.