Open AccessSums of Powers of Integers and Bernoulli Numbers Clarified
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.v9n2p12
Subject(s) - bernoulli number , bernoulli's principle , inverse , combinatorics , pascal (unit) , physics , matrix (chemical analysis) , mathematics , discrete mathematics , quantum mechanics , geometry , materials science , composite material , thermodynamics
This work exposes a very simple method for calculating at the same time the sums of powers of the first integers S_m(n) and the Bernoulli numbers B_m. This is possible thank only to the relation S_m(x+1)-S_m(x)= x^m and the Pascal formula concerning S_m(n) which may be explained as if the vector n^2-n, n^3-n,...,n^(m+1)-n is the transform of the vector S_1(n), S_2(n),...,S_m(n) by a matrix P built from the Pascal triangle. Very useful relations between the sums S_m(n), the Bernoulli numbers B_m and elements of the inverse matrix of P are deduced, leading straightforwardly to known and new properties of them.