  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.
