Sums of Powers of Integers and Bernoulli Numbers Clarified

DO TAN SI

Abstract


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.


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DOI: https://doi.org/10.5539/apr.v9n2p12

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