Abstract
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.
Highlights
IntroductionIn an article published on the last decade Chen, Fu and Zhang (2009) showed that sums of powers on arithmetic progression may be deduced from the original Faulhaber theorem or calculated by utilizing the central factorial numbers as in the approach of Knuth (1993)
Because Sm (z, n) and the Bernoulli polynomials Bm (z) are differential transforms of monomials, we will see that they satisfy the addition formula which in turn leads to formulae allowing the calculations of Sm (z, n) from sums of powers of integers Sm (n) = Sm (0, n) as so as Bm (z) from Bernoulli numbers Bm = Bm (0)
The main remark about this work is that it steadily utilizes the operator calculus method for obtaining a simple algorithm for calculating the Bernoulli numbers and sums of powers of integers leading to the obtention of Bernoulli polynomials and sums of powers on arithmetic progressions by the theorem “ In Sm( z,n ), the coefficients of zk is for k = 0,1,..., m ; the coefficient of nk is for k = 1, 2,..., m +1
Summary
In an article published on the last decade Chen, Fu and Zhang (2009) showed that sums of powers on arithmetic progression may be deduced from the original Faulhaber theorem or calculated by utilizing the central factorial numbers as in the approach of Knuth (1993). The methods utilized by Faulhaber, Knuth and these authors are respectful but the resulted formulae are not so easy to get and apply, especially for high value of powers. Searching for another and simpler method for obtaining Sm (z, n) , we observe firstly that it is the transform of the monomial zm by a differential operator built from the derivative operator Dz. Sm (z, n) = We propose as applications the calculations of alternate sums of powers of integers and a criteria for a polynomials to have only integer values
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
