Sums of powers of integers—A review

Abstract

The problem of finding formulae for the sums of powers of integers has been tackled by various methods throughout the literature. A review of the main techniques is given with emphasis on both the expanded polynomial forms and the more familiar compact factored forms of the formulae. For large powers, the factored forms can be obtained more readily by first deriving a special well‐defined polynomial form called the Faulhaber form. Most of the material on algorithms for generating the coefficients of the Falhauber polynomials is new. The expanded forms of the formulae involve Bernoulli numbers which are important in other applications of interest to mathematics and statistics majors.