Obtaining Concise Formulae on Bernoulli Polynomials-Numbers and Sums of Powers-Faulhaber Problems

Abstract

Utilizing the translation operator exp (a\(\partial\)_z)  to represent Bernoulli polynomials B_m(z) and power sums  S_m(z,n) as polynomials of Appell-type , we obtain concisely almost all their known properties as so as many new ones, especially very simple symbolic formulae for calculating Bernoulli numbers and polynomials, power sums of entire and complex numbers. Then by the change of arguments from z into  Z = z(z-1) and n into \(\lambda\) which is the 1^st order power sum we obtain the Faulhaber formula for powers sums  in term of polynomials in \(\lambda\) having coefficients depending on Z. Practically we give tables for calculating in easiest possible manners Tables of Bernoulli numbers, Bernoulli polynomials, sums of powers of complex numbers are given.