TITLE:
The Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked
AUTHORS:
Do Tan Si
KEYWORDS:
Bernoulli Numbers, Bernoulli Polynomials, Powers Sums, Faulhaber Conjecture, Shift Operator, Operator Calculus
JOURNAL NAME:
Applied Mathematics,
Vol.10 No.3,
March
22,
2019
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.