On Sylvester Waves and restricted partitions

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.