Sums of Powers and Harmonic Numbers: A new approach

Abstract

There have been derivations for the Sums of Powers published since the sixteenth century. All techniques have used recursive processes, producing the following formula in the series. I present a new method that calculates the Sums of Powers and Harmonic Numbers. Starting with a novel relationship between Pascal’s Numbers and Stirling’s Numbers of the First Kind, the Sums of Powers is developed. This formula, published previously using a different methodology, is in terms of Pascal Numbers multiplied by constant coefficients. However, a further step is introduced. A recursive relationship is discovered among the coefficients of these formulae. A double sigma master formula is developed, allowing one to calculate all formulae for Sums of Powers without needing Bernoulli Numbers. Finally, from the Sums of Powers master formula, I derive a formula to calculate the Bernoulli Numbers. I further develop a summation formula for the Harmonic Numbers using the same relationships.