Lower Bounds for Densities of Transcendental Gamma-Function Derivatives

Abstract

In recent work, we showed that for all $q\in \frac{1}{2}\mathbb{Z}\setminus {\mathbb{Z}}_{\le 0}$ the sequence ${\left\{{\Gamma }^{\left(n\right)}\left(q\right)\right\}}_{n\ge 1}$ contains transcendental elements infinitely often, with the density of transcendental ${\Gamma }^{\left(n\right)}\left(q\right)$ among $n\in \left\{1,2,\dots ,N\right\}$ 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 $q=m\in \left\{1,2,\dots \right\}$ and rationally shifted lattice points $q=\tilde{m}\in \left\{\kappa ,\pm 1+\kappa ,\pm 2+\kappa ,\dots \right\}$ (for $\kappa \in \left(0,1\right)\cap \mathbb{Q}$ such that $\Gamma \left(\kappa \right)$ is transcendental). For $n\in {\mathbb{Z}}_{\ge 2}$, we find there are at most $n-1$ algebraic ${\Gamma }^{\left(n\right)}\left(m\right)$, and for $n\in {\mathbb{Z}}_{\ge 1}$, there are at most $n$ algebraic ${\Gamma }^{\left(n\right)}\left(\tilde{m}\right)$ for each one-sided shifted lattice (i.e., $\tilde{m}\ge \kappa$ or $\tilde{m}\le \kappa$). These results form the basis for constructing lower bounds for the densities of transcendental ${\Gamma }^{\left(n\right)}\left(m\right)$ among $m\in \left\{1,2,\dots ,M\right\}$ and transcendental ${\Gamma }^{\left(n\right)}\left(\tilde{m}\right)$ among either $\tilde{m}\in \left\{\kappa ,1+\kappa ,2+\kappa ,\dots ,M+\kappa \right\}$ or $\tilde{m}\in \left\{\kappa ,-1+\kappa ,-2+\kappa ,\dots ,-M+\kappa \right\}$. Allowing $n$ to vary, we derive lower bounds for the bivariate densities of both transcendental ${\Gamma }^{\left(n\right)}\left(m\right)$ among $\left(n,m\right)\in \left\{2,3,\dots ,N\right\}\times \left\{1,2,\dots ,M\right\}$ and transcendental ${\Gamma }^{\left(n\right)}\left(\tilde{m}\right)$ among one-sided shifted-lattice analogues.