Obtaining Easily Powers Sums on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus

We show that a sum of powers on an arithmetic progression is the transform of a monomial by a differential operator and that its generating function is simply related to that of the Bernoulli polynomials from which consequently it may

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The Faulhaber Conjecture Resolved Generalization to Powers Sums on Arithmetic Progressions

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

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