Alternate Sylvester sums on the Frobenius set

The alternate Sylvester sums are Tm(a,b)=∑n∈NR(−1)nnmTm(a,b)=∑n∈NR(−1)nnm, where aa and bb are coprime, positive integers, and NRNR is the Frobenius set associated with aa and bb. In this note, we study the generating functions, recurrences and explicit expressions of the alternate Sylvester sums. It can be found that the results are closely related to the Bernoulli polynomials, the Euler polynomials, and the (alternate) power sums over the natural numbers.