A type of multiple integral with loggamma function

Abstract

In this paper, we study the multiple integral $I={\int }_0^1{\int }_0^1\cdots {\int }_0^{1}f(x_1+x_2+\cdots +x_n)d{x}_{1}d{x}_2\dots d{x}_n$. A general formula of $I$ is presented. As an application, the integral $I$ with $f(x)=\log \Gamma (x)$ is evaluated. We show that the values of $I$ share a common formula for all $n\in \mathbb{N}$. The subsidiary computational challenges are substantial and interesting in their own right.

Submitted to Involve