Gautchi's ratio and the Volume of the unit ball in R^n

TL;DR

This paper explores the properties of the volume of the unit ball in n-dimensional space using gamma function ratios, deriving formulas and inequalities that deepen understanding of geometric volume relationships.

Contribution

It introduces a novel infinite product formulation of the gamma function ratio to analyze the volume ratios of unit balls in different dimensions.

Findings

Derived new formulas for volume ratios of unit balls in R^n

Established inequalities involving gamma function ratios and volumes

Reproduced known and novel relationships between volumes in different dimensions

Abstract

Let Omega(n) be the volume of the unit ball in R^n. We formulate as an infinite product the gamma function ratio gamma(x+1/2)/gamma(x),x>0, which allows us to reproduce and /or produce a variety of formulas and inequalities, some of them seemingly new, concerning Omega(n-1)/Omega(n),and (Omega(n))^2/Omega(n-1)Omega(n+1)