"Infinitely Often" Transcendence of Gamma-Function Derivatives
Michael R. Powers
TL;DR
The paper investigates the arithmetic nature of Gamma-function derivatives $\Gamma^{(n)}(q)$ at rational arguments $q\in\mathbb{Q}\setminus\mathbb{Z}_{\le0}$ and extends known infinitely-often transcendence results from $q=1$ to all half-integers $q\in\tfrac{1}{2}\mathbb{Z}\setminus\mathbb{Z}_{\le0}$. For rational $q$ not in $\tfrac{1}{2}\mathbb{Z}$, it proves a disjunctive result: at least one of $\{\Gamma^{(n)}(q)\}$ or $\{\Gamma^{(n)}(1-q)\}$ contains infinitely many transcendental elements. The approach hinges on expressing products like $\Gamma(q+t)\Gamma(q-t)$ and $\Gamma(q+t)\Gamma(1-q-t)$ via the Gamma functional equation and Euler's reflection formula, obtaining series with coefficients tied to algebraic numbers and powers of $\pi$, and deriving a contradiction from the (assumed) finite-dimensional span of these coefficients together with the transcendence of $\pi$. These results extend unconditionally known transcendence phenomena for gamma-derivatives and pave the way for deeper investigations into the arithmetic of special-function 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 elements of the sequence $\left\{ Γ^{\left(n\right)}\left(1\right)\right\} _{n\geq1}$ are transcendental infinitely often. This result is now generalized to all sequences $\left\{ Γ^{\left(n\right)}\left(q\right)\right\} _{n\geq1}$ for $q\in\tfrac{1}{2}\mathbb{Z}\setminus\mathbb{Z}_{\leq0}$. Furthermore, for $q\in\mathbb{Q}\setminus\tfrac{1}{2}\mathbb{Z}$ we show that at least one of the sequences $\left\{ Γ^{\left(n\right)}\left(q\right)\right\} _{n\geq1}$, $\left\{ Γ^{\left(n\right)}\left(1-q\right)\right\} _{n\geq1}$ contains infinitely many transcendental elements.