Faulhaber’s polynomials, Bernoulli’s numbers, and Conway’s inductive procedure

Abstract:

Johann Faulhaber, the "Great Calculator of Ulm", in 1631 published explicit formulas for the sums of powers of natural numbers 1k + 2k + + nk as polynomials in n, for k = 1 to 17.

He also discovered remarkable properties of these polynomials, including the first use of the Derivative in mathematics: curiously separate from any connection to slopes of tangent lines.

About 100 years later appeared Jacob Bernoulli's general formula for Faulhaber's polynomials in terms of what are now called Bernoulli numbers, and which appear in many areas of mathematics, from the zeta function to combinatorics.

In this talk we will give a self-contained derivation of some of these ideas using first year linear algebra and an inductive approach of John Conway. This allows us to calculate successive Faulhaber polynomials and Bernoulli numbers in a direct and simple fashion.