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 S a , b k ( x ) which is given by S a , b k ( x ) = b k + ( a + b ) k + ( 2 a + b ) k + ⋯ + ( a ( x − 1 ) + b ) k .