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

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.

arXiv comment7 pages, 1 figure. Fully revised and shortened (02/05/2014)

References in corpus (1)

• A New Approach to Rational Values of Trigonometric Functions