Faulhaber's Theorem on Power Sums

We observe that the classical Faulhaber's theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression $a+b, a+2b, ..., a+nb$ is a polynomial in $na+n(n+1)b/2$. While this assertion can be deduced from the original Fauhalber's theorem, we give an alternative formula in terms of the Bernoulli polynomials. Moreover, by utilizing the central factorial numbers as in the approach of Knuth, we derive formulas for $r$-fold sums of powers without resorting to the notion of $r$-reflexive functions. We also provide formulas for the $r$-fold alternating sums of powers in terms of Euler polynomials.Comment: 12 pages, revised version, to appear in Discrete Mathematic

Sums of Powers and the Bernoulli Numbers

This expository thesis examines the relationship between finite sums of powers and a sequence of numbers known as the Bernoulli numbers. It presents significant historical events tracing the discovery of formulas for finite sums of powers of integers, the discovery of a single formula by Jacob Bernoulli which gives the Bernoulli numbers, and important discoveries related to the Bernoulli numbers. A method of generating the sequence by means of a number theoretic recursive formula is given. Also given is an application of matrix theory to find a relation, first given by Johannes Faulhaber, between finite sums of odd powers and finite sums of even powers. An approach to finding a formula for sums of powers using integral calculus is also presented. The relation between the Bernoulli numbers and the coefficients of the Maclaurin expansion of f(z) = z /ez - 1, which was first given by Léonard Euler, is considered, as well as the trigonometric series expansions which are derived from the Maclaurin expansion of f(z), and the zeta function. Further areas of research relating to the topic are explored

Sums of Powers and the Bernoulli Numbers

This expository thesis examines the relationship between finite sums of powers and a sequence of numbers known as the Bernoulli numbers. It presents significant historical events tracing the discovery of formulas for finite sums of powers of integers, the discovery of a single formula by Jacob Bernoulli which gives the Bernoulli numbers, and important discoveries related to the Bernoulli numbers. A method of generating the sequence by means of a number theoretic recursive formula is given. Also given is an application of matrix theory to find a relation, first given by Johannes Faulhaber, between finite sums of odd powers and finite sums of even powers. An approach to finding a formula for sums of powers using integral calculus is also presented. The relation between the Bernoulli numbers and the coefficients of the Maclaurin expansion of f(z) = z /ez - 1, which was first given by Léonard Euler, is considered, as well as the trigonometric series expansions which are derived from the Maclaurin expansion of f(z), and the zeta function. Further areas of research relating to the topic are explored