The group generated by the gamma functions $\Gamma(ax+1)$, and its subgroup of the elements converging to constants

Abstract

Let $G$ be the multiplicative group generated by the gamma functions $\Gamma(ax+1)$ $(a=1,2,\dots)$, and $H$ be the subgroup of all elements of $G$ that converge to nonzero constants as $x\rightarrow\infty$. The quotient group $G/H$ is the group of equivalence classes of $G$, where $f$ and $g$ are equivalent $\iff f\sim Cg$ $(x\rightarrow\infty)$ for some $C\not=0$. We show that $G/H\simeq\mathbb{Q}^+$. A similar consideration is possible for the case that the gamma functions $\Gamma(ax+1)$ with $a\in\mathbb{R}^+$ are concerned, and we show that $G/H\simeq\mathbb{Z}\times\mathbb{R}\times\mathbb{R}$. Also, several concrete examples of the elements of $H$ are constructed, e.g., it holds that $\frac{\binom{18n}{12n,3n,3n}}{\binom{18n}{9n,8n,n}}\longrightarrow\sqrt{\frac{2}{3}}$ $(n\rightarrow\infty)$, where $\binom{*}{*,\dots,*}$ denotes a multinomial coefficient.