Open AccessObtaining Easily Sums of Powers on Arithmetic Progressions and Properties of Bernoulli Polynomials by Operator Calculus
Author(s) -
Do Tan Si
Publication year - 2017
Publication title -
applied physics research
Language(s) - English
Resource type - Journals
eISSN - 1916-9647
pISSN - 1916-9639
DOI - 10.5539/apr.v9n5p73
Subject(s) - monomial , bernoulli's principle , operator (biology) , mathematics , bernoulli polynomials , bernoulli number , simple (philosophy) , bernoulli process , generating function , bernoulli distribution , differential (mechanical device) , function (biology) , differential operator , algebra over a field , discrete mathematics , pure mathematics , difference polynomials , orthogonal polynomials , random variable , philosophy , repressor , aerospace engineering , chemistry , engineering , biology , biochemistry , epistemology , evolutionary biology , transcription factor , statistics , gene
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.