Alternate Sylvester sums on the Frobenius set

The alternate Sylvester sums are Tm(a,b)=∑n NR(−1)nnm, where a and b are coprime, positive integers, and NR is the Frobenius set associated with a and b. 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.