A refinement of Faulhaber’s theorem concerning sums of powers of natural numbers

In an attempt to present a refinement of Faulhaber’s theorem concerning sums of powers of natural numbers, the authors investigate and derive all the possible decompositions of the polynomial Sa,bk(x) which is given by Sa,bk(x)=bk+(a+b)k+(2a+b)k+⋯+(a(x−1)+b)k.