On the possible exceptions for the transcendence of the log-gamma function at rational entries

TL;DR

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Contribution

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Abstract

In a recent work [JNT \textbf{129}, 2154 (2009)], Gun and co-workers have claimed that the number $\,\log{\Gamma(x)} + \log{\Gamma(1-x)}\,$, $x$ being a rational number between $0$ and $1$, is transcendental with at most \emph{one} possible exception, but the proof presented there in that work is \emph{incorrect}. Here in this paper, I point out the mistake they committed and I present a theorem that establishes the transcendence of those numbers with at most \emph{two} possible exceptions. As a consequence, I make use of the reflection property of this function to establish a criteria for the transcendence of $\,\log{\pi}$, a number whose irrationality is not proved yet. This has an interesting consequence for the transcendence of the product $\,\pi \cdot e$, another number whose irrationality remains unproven.