Hyperelliptic values of the Gamma function

TL;DR

This paper presents a method to compute specific Gamma function values at rational points in (0,1) using hyperelliptic integrals and transcendental constants, bypassing elliptic functions.

Contribution

It introduces a novel approach to evaluate Gamma function values at rational arguments using hyperelliptic integrals, avoiding elliptic K-functions.

Findings

Explicit formulas for Gamma values at certain rational points

Use of hyperelliptic integrals and transcendental constants

Simplification of calculations without elliptic functions

Abstract

We show how to calculate particular values of the Gamma function for specific rational arguments in the interval (0,1) without using the Elliptic K-function. Instead we use transcendental constants or periods defined by hyperelliptic integrals.