The Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked

Do Tan Si

doi:10.4236/am.2019.103009

Abstract

Utilizing translation operators we get the powers sums on arithmetic progressions and the Bernoulli polynomials of order munder the form of differential operators acting on monomials. It follows that (d/dn-d/dz) applied on a power sum has a meaning and is exactly equal to the Bernoulli polynomial of the same order. From this new property we get the formula giving powers sums in term of sums of successive derivatives of Bernoulli polynomial multiplied withprimitives of the same order of n. Then by changing the two arguments z,n into Z=z(z-1), λ where λ designed the 1st order power sums and proving that Bernoulli polynomials of odd order vanish for arguments equal to 0, 1/2, 1, we obtain easily the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. These coefficients are found to be derivatives of odd powers sums on integers expressed in Z. By the way we obtain the link between Faulhaber formulae for powers sums on integers and on arithmetic progressions. To complete the work we propose tables for calculating in easiest manners possibly the Bernoulli numbers, the Bernoulli polynomials, the powers sums and the Faulhaber formula for powers sums.

Highlights

By changing the two arguments z, n into =Z z ( z −1), λ where λ designed the 1st order power sums and proving that Bernoulli polynomials of odd order vanish for arguments equal to 0, 1/2, 1, we obtain the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z
The main particularity of this work consists in obtaining Sm ( z,n) as the transform of zm by a differential operator, as so as the Bernoulli polynomial Bm ( z), from which we deduce the new formula (∂n − ∂z ) Sm ( z, n) = Bm ( z) and get immediately Sm ( z, n) as polynomials in n
From which we get the property saying that B1−1 ( z ) B2k+1 ( z ) is a homogeneous polynomial of order k in =Z z ( z −1)

Summary

Introduction

It is plausible that from this list he observed that ∑ n m may be written in terms of the numbers Bk which are the same for all m This famous conjectured formula of Bernoulli was proven in 1755 based on the calculus of finite difference by Euler [7], a researcher working with the Bernoulli brothers at Zurich. We know that Jacobi [9] has the merit of giving the right proof for this conjecture and calculating the first six Faulhaber coefficients A(jm) he did not get a formula for obtaining all of them. Another merit of Jacobi consists in pioneering the use of the derivative with respect to n when observing that. From (2.18) we see that Bm (0) are identifiable with Bernoulli numbers Bm

The Powers Sums

Calculations of Bernoulli Numbers

Calculations of Bernoulli Polynomials

Calculations of Faulhaber Powers Sums

Remarks and Conclusions