Lower Bounds for Densities of Transcendental Gamma-Function Derivatives

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.