The Vandermonde Connection

Divided differences are the building blocks of polynomial interpolation — the coefficients that generalize finite differences to unequally spaced data. Multiple integrals involving Vandermonde determinants appear in random matrix theory, statistical mechanics, and multivariate approximation. These are different subjects with different motivations.

Floater proves they’re connected: a class of multiple integrals involving Vandermonde polynomials equals a divided difference expression.

The identity works in both directions. Read one way, it gives an integral formula for divided differences — expressing these discrete, combinatorial objects as continuous integrals over simplices weighted by Vandermonde factors. Read the other way, it evaluates multidimensional integrals by reducing them to divided differences — which are computable by simple recurrence.

Along the way, the paper proves that both pure and mixed partial derivative sums of Vandermonde polynomials vanish. These vanishing results — which the author suggests have independent significance — are the technical engine driving the identity. The Vandermonde determinant’s algebraic structure forces cancellations that make the integral collapse to the divided difference.

The connection is not between two different mathematical objects but between two views of the same one. Divided differences already encode integral information through their definition as limits of difference quotients. The identity makes this implicit connection explicit and computable, bridging discrete interpolation theory and continuous integral calculus through the Vandermonde determinant.