Author(s)

Do Tan Si

The HoChiMinh-City Physical Association, Ho Chi Minh City, Vietnam.

Utilizing the translation operator to represent Bernoulli polynomials and power sums as polynomials of Sheffer-type, we obtain concisely almost all their known properties as so as many new ones, especially new recursion relations for calculating Bernoulli polynomials and numbers, new formulae for obtaining power sums of entire and complex numbers. Then by the change of arguments from z into Z = z(z-1) and n into λ which is the 1^st order power sum we obtain the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. Practically we give tables for calculating in easiest possible manners, the Bernoulli numbers, polynomials, the general powers sums.

Bernoulli Numbers, Bernoulli Polynomials, Powers Sums, Zeta Function, Faulhaber Conjecture

Si, D. (2022) Selection of Coherent and Concise Formulae on Bernoulli Polynomials-Numbers-Series and Power Sums-Faulhaber Problems. Applied Mathematics, 13, 799-821. doi: 10.4236/am.2022.1310051.