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

TL;DR

This paper investigates the structure of the group generated by gamma functions and its subgroup of elements converging to constants, revealing isomorphisms with rational and real product groups, and provides concrete examples of such elements.

Contribution

It characterizes the quotient group of gamma function-generated groups and their subgroups, establishing isomorphisms with well-known algebraic structures and constructing explicit examples.

Findings

The quotient group G/H is isomorphic to + for gamma functions with integer parameters.

For real positive parameters, G/H is isomorphic to   .

Explicit examples of elements in H show convergence to specific constants, such as /3.

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.