On the Sylvester waves in partition function

TL;DR

This paper provides a formal proof for a new explicit expression of Sylvester waves in the partition function, representing them as sums over Bernoulli polynomials with periodic coefficients, enhancing understanding of partition function decomposition.

Contribution

It introduces a formal proof validating a new representation of Sylvester waves as weighted sums over Bernoulli polynomials with shifted arguments.

Findings

Explicit expression of Sylvester waves as finite sums over Bernoulli polynomials.

Validation of the representation as a sum of polynomial terms with shifted arguments.

Enhanced theoretical understanding of the structure of partition functions.

Abstract

Sylvester showed that the partition function can be written as a sum of the polynomial term and quasiperiodic components called the Sylvester waves. Recently an explicit expression of the Sylvester wave as a finite sum over the Bernoulli polynomials of higher order with periodic coefficients was found. This expression can be also written as the weighted sum of the polynomial terms with shifted arguments and this manuscript presents a formal proof for validity of such representation.