Evaluation of multivariate integrals based on a duality identity for the Stieltjes transform

TL;DR

This paper investigates a duality identity for the Stieltjes transform derived from a double integral representation of Catalan's constant, extending it to multiple dimensions and linking it to multivariate zeta functions.

Contribution

It introduces a new duality identity for the Stieltjes transform based on integral representations and extends this to higher dimensions, connecting to multivariate zeta functions.

Findings

Derived a duality identity for the Stieltjes transform

Extended the duality to arbitrary dimensions

Linked the results to multivariate zeta functions

Abstract

A detailed study of a double integral representation of the Catalan's constant allows us to identify a duality identity for the Stieltjes transform on which it is based. This duality identity is then extended to an arbitrary dimensional integral and several special cases are deduced. On the way, we also highlight a relationship with some multivariate generalizations of the Riemann zeta function.