Lower Bounds for Densities of Transcendental Gamma-Function Derivatives

TL;DR

This work investigates how often derivatives of the Gamma function at integer arguments are transcendental, analyzing three counting regimes: fixed $m$ with $n\le N$, fixed $n$ with $m\le M$, and the pair set $(n,m)$. The authors leverage a generating function approach, a linear relation expressing $\Gamma^{(n)}(m)$ in terms of $\Gamma^{(\ell)}(1)$ with a rational coefficient matrix, and determinant-based invertibility arguments to bound the number of algebraic values, yielding unconditional density bounds. Key results include a lower bound $\delta_m(N)\ge \beta_m(N)=\max\{0,\sqrt{N-1}-3/2\}/N$ for fixed $m$, a strong bound showing at most $n-1$ algebraic values among $m$-samples for fixed $n$, and a corollary giving $\delta_n(M)\ge \beta_n(M)=1-\min\{n-1,M\}/M$; collectively, these imply a bivariate density bound $\delta(N,M)\ge \beta(N,M)$ with different regimes depending on $M$ relative to $N$. The results underscore that the density is highly regime-dependent, approaching 1 when $M$ grows with $N$, while offering a path toward improving the fixed-$m$ bound via deeper interplay between the $n$ and $m$ dimensions. Overall, the paper advances unconditional understanding of the arithmetic nature of Gamma-derivative values and their transcendence densities. All results are stated with explicit asymptotics and carefully constructed algebraic- transcendental arguments.

Abstract

In recent work, we showed that for all $q\in\tfrac{1}{2}\mathbb{Z}\setminus\mathbb{Z}_{\leq0}$, the sequence $\left\{Γ^{\left(n\right)}\left(q\right)\right\}_{n\geq1}$ contains transcendental elements infinitely often. Employing methods from that analysis, we now find that for arbitrary fixed $m\in\mathbb{Z}_{\geq1}$, the density of the set of transcendental $Γ^{\left(n\right)}\left(m\right)$ for $n\in\left\{1,2,\ldots,N\right\}$ is bounded below by $β_{m}\left(N\right)=\max\left\{ 0,\sqrt{N-1}-3/2\right\}/N$, so that $β_{m}\left(N\right)\asymp N^{-1/2}\rightarrow0$ as $N\rightarrow\infty$. Fixing $n\in\mathbb{Z}_{\geq2}$, we further provide a relatively strong upper bound (of $n-1$) on the number of algebraic $Γ^{\left(n\right)}\left(m\right)$ for $m\in\mathbb{Z}_{\geq1}$, and then show that this result implies that the density of the set of transcendental elements $Γ^{\left(n\right)}\left(m\right)$ for $m\in\left\{1,2,\ldots,M\right\}$ has lower bound $β_{n}\left(M\right)=1-\min\left\{ n-1,M\right\}/M$, with $β_{n}\left(M\right)\asymp1-\left(n-1\right)/M\rightarrow1$ as $M\rightarrow\infty$. Finally, we evaluate the bivariate density of the set of transcendental $Γ^{\left(n\right)}\left(m\right)$ for $\left(n,m\right)\in\left\{2,3,\ldots,N\right\}\times\left\{1,2,\ldots,M\right\}$, obtaining the lower bound $β\left(N,M\right)=\left(M-1\right)/\left[2\left(N-1\right)\right]$ if $M\leq N-1$, $=1-N/\left(2M\right)$ if $M>N-1$, which converges to 0 for $M/N\rightarrow0$, to 1/2 for $M/N\rightarrow1$, and to 1 for $M/N\rightarrow\infty$.

Theorems & Definitions (12)

• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 2
• proof
• Corollary 1
• proof