Integral representations for local dilogarithm and trilogarithm functions

TL;DR

This paper introduces new integral representations for dilogarithm and trilogarithm functions, leading to novel formulas for constants and functions, as well as bounds and Euler sums, enriching the mathematical understanding of these special functions.

Contribution

It provides new integral formulas for dilogarithm and trilogarithm functions, and derives related representations for constants, bounds, and Euler sums, advancing the theoretical framework of special functions.

Findings

New integral representations for dilogarithm and trilogarithm functions

Derived formulas for Apery, Catalan constants, and Legendre chi functions

Established a lower bound for the dilogarithm function

Abstract

We show new integral representations for dilogarithm and trilogarithm functions on the unit interval. As a consequence, we also prove (1) new integral representations for Apery, Catalan constants, and Legendre chi functions of order 2, 3, (2) a lower bound for the dilogarithm function on the unit interval, (3) new Euler sums.