Obtaining Concise Formulae on Bernoulli Polynomials-Numbers and Sums of Powers-Faulhaber Problems | Chapter 2 | Research Highlights in Mathematics and Computer Science Vol. 6

 Utilizing the interpretation operator exp (aμz)  to show Bernoulli polynomials Bm(z) and power sums  Sm(z,n) as polynomials of Appell-type , we acquire concisely principal part their known properties as so as many new one, especially very natural symbolic formulae for wily Bernoulli numbers and polynomials, power sums of whole and complex numbers. Then by the change of debates from z into  Z = z(z-1) and n into λ that is the 1st order power total we obtain the Faulhaber recipe for powers sums  in term of polynomials in λ having coefficients contingent upon Z. Practically we give tables for wily in easiest attainable manners Tables of Bernoulli numbers, Bernoulli polynomials, sums of capacities of complex numbers are given.

Author(s) Details:

Do Tan Si,
The HoChiMinh-City Physical Association, Vietnam and ULB and UEM, Belgium.

Please see the link here: https://stm.bookpi.org/RHMCS-V6/article/view/9677

Obtaining Easily Powers Sums on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus | Chapter 7 | New Insights into Physical Science Vol.3

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 be calculated. Besides, we show that it is obtainable also from the sums of powers of integers, i.e. from the Bernoulli numbers which in turn may be calculated by a simple algorithm.  By the way, for didactic purpose, operator calculus is utilized for proving in a concise manner the main properties of the Bernoulli polynomials.

Author(s) Details

Do Tan Si HoChiMinh-City
Physical Association, Vietnam and ULB and UEM, Belgium.

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