对数Γ函数在有理元处超越的可能例外
On the possible exceptions for the transcendence of the log-gamma function at rational entries
摘要：
在最近的工作[JNT \textbf{129}，2154（2009）]中，Gun和同事声称数$\,\log{Γ(x)} + \log{Γ(1-x)}\,$，$x$是介于$0$和$1$之间的有理数，是超越的，至多有\emph{one}可能的例外，但在该工作中给出的证明是\emph{incorrect}。在本文中，我指出了他们犯的错误，并提出了一个定理，该定理建立了这些数的超越性，最多有\emph{two}可能的例外。因此，我利用这个函数的反射性质建立了一个超越$\,\logπ$的准则，这个数的非理性还没有得到证明。这对乘积$\,π\cdot e$的超越性有一个有趣的结果，这是另一个其不合理性尚未被证明的数。
英文摘要：
In a recent work [JNT \textbf{129}, 2154 (2009)], Gun and co-workers have claimed that the number $\,\log{Γ(x)} + \log{Γ(1-x)}\,$, $x$ 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 $\,\logπ$, a number whose irrationality is not proved yet. This has an interesting consequence for the transcendence of the product $\,π\cdot e$, another number whose irrationality remains unproven.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
