Formulas for sums of powers of integers by functional equations

Abstract

This paper presents a direct and simple approach to obtaining the formulas forS k(n)= 1 k + 2 k + ... +n k wheren andk are nonnegative integers. A functional equation is written based on the functional properties ofS k (n) and several methods of solution are presented. These lead to several recurrence relations for the functions and a simple one-step differential-recurrence relation from which the polynomials can easily be computed successively. Arbitrary constants which arise are (almost) the Bernoulli numbers when evaluated and identities for these modified Bernoulli numbers are obtained. The functional equation for the formulas leads to another functional equation for the generating function for these formulas and this is used to obtain the generating functions for theS k 's and for the modified Bernoulli numbers. This leads to an explicit representation, not a recurrence relation, for the modified Bernoulli numbers which then yields an explicit formula for eachS k not depending on the earlier ones. This functional equation approach has been and can be applied to more general types of arithmetic sequences and many other types of combinatorial functions, sequences, and problems.