  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.
