A divided difference identity for a class of multiple integrals

TL;DR

This paper introduces a novel identity linking multiple integrals with Vandermonde polynomials to divided differences, providing a new integral formula and revealing properties of derivatives of Vandermonde polynomials.

Contribution

The paper presents a new identity connecting multiple integrals and divided differences, along with insights into derivatives of Vandermonde polynomials.

Findings

Derived an identity relating multiple integrals and divided differences

Showed sums of derivatives of Vandermonde polynomials are zero

Provided an integral formula for divided differences

Abstract

We derive an identity that relates a class of multiple integrals involving Vandermonde polynomials to divided differences. Alternatively the identity can be viewed as an integral formula for divided differences. As part of the derivation we show that both sums of pure partial derivatives and mixed partial derivatives of Vandermonde polynomials are zero, which might be of independent interest.