A curious formula related to the Euler Gamma function

TL;DR

This paper presents a new formula expressing the logarithm of the Gamma function for x in (0,1) using an infinite series and special constants, revealing a novel relationship involving the Euler Gamma function.

Contribution

It introduces a new explicit formula for log Gamma(x) involving an infinite sine series and a special constant, expanding understanding of Gamma function representations.

Findings

Derived a new series representation for log Gamma(x)

Identified the constant eta with high precision

Connected Gamma function properties with sine series

Abstract

In this note, we prove that for all $x \in (0 , 1)$, we have: $$ \log\Gamma(x) = \frac{1}{2} \log\pi + \pi \boldsymbol{\eta} \left(\frac{1}{2} - x\right) - \frac{1}{2} \log\sin(\pi x) + \frac{1}{\pi} \sum_{n = 1}^{\infty} \frac{\log n}{n} \sin(2 \pi n x) , $$ where $\Gamma$ denotes the Euler Gamma function and $$ \boldsymbol{\eta} := 2 \int_{0}^{1} \log\Gamma(x) \cdot \sin(2 \pi x) \, d x = 0.7687478924\dots $$