A product of Gamma function values at fractions with the same denominator

TL;DR

This paper evaluates the product of Gamma function values at fractions with a fixed denominator, revealing that the result depends on whether the denominator is a prime power, thus connecting number theory and special functions.

Contribution

It provides a closed-form evaluation of the Gamma function product over fractions with a given denominator, highlighting the role of prime powers in the result.

Findings

Product depends on whether the denominator is a prime power

Explicit formulas derived for the Gamma function products

Links number theory with special function evaluations

Abstract

Let D(n) be the set of all fractions in the unit interval whose denominator in lowest terms equals $n$. We evaluate the product of the values of the Gamma function at the points of D(n), as a function of $n$; the answer depends on whether or not $n$ is a prime power.