On Sylvester Waves and restricted partitions

TL;DR

This paper explores the properties of Sylvester waves and restricted partitions using finite difference operators, deriving new expressions for Euler's functions, Todd operator, and connecting Dedekind sums with historical proofs.

Contribution

It introduces a general form for Sylvester waves via multiplicative series and discusses their reduction, linking classical results with modern spectral and difference operator techniques.

Findings

Derived elegant expressions for Euler's rational functions and Todd operator.

Connected Dedekind sums with early proofs by Brioschi.

Presented a spectral perspective on finite difference properties of Riesz means.

Abstract

The higher Sylvester waves are discussed. Techniques used involve finite difference operators. For example, using Herschel's theorem, elegant expressions for Euler's rational functions and the Todd operator are found. Derivative expansions are also rapidly treated by the same method. A general form for the wave is obtained using multiplicative series, and comments made on its further reduction. As is known, Dedekind sums arise in the case of coprime components and it is pointed out that Brioschi had this result, but not the terminology, very early on. His proof is repeated. Adding a set of ones to the components of the denumerant corresponds to a succession of (discrete) smoothings and is a Cesaro sum. Using a spectral vocabulary, I take the opportunity to exhibit the finite difference counterparts of some continuous, distributional properties of Riesz typical means.