Restricted partition as Bernoulli polynomials of higher orderAlgebra & Discrete Mathematics
|Boris Rubinstein, UC Davis
Explicit expressions for restricted partition function (Sylvester's denumerant) and its quasiperiodic components (called Sylvester waves) for a set of positive integers are derived. I show an explicit formula for the Sylvester wave in a form of finite sum of the Bernoulli polynomials of higher order multiplied by a periodic function of integer period. The periodic factor is expressed through the Eulerian polynomials of higher order (introduced by L.Carlitz). I show that it is possible to represent this result also in a form of a finite sum over Bernoulli polynomials of higher order with periodic coefficients.