On the Sylvester waves in partition function

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.