The Faulhaber Conjecture Resolved Generalization to Powers Sums on Arithmetic Progressions

Abstract

By comparing the formula giving odd powers sums of integers from Bernoulli numbers and the Faulhaber conjecture form of them, we obtain two recurrence relations for calculating the Faulhaber coefficients. Parallelly we search for and obtain the differential operator which transform a powers sum into a Bernoulli polynomial. From this and by changing arguments from z,n into Z=z(z-1), ?=zn+n(n-1/2) we obtain a formula giving powers sums on arithmetic progressions directly from the  powers sums on integers.