The formal Derivative and Integral, and Faulhaber’s polynomials of the second kind

Abstract:

The Derivative arguably first appeared in mathematics in the 1631 work of Johann Faulhaber, who used it to establish remarkable relations between sums of kth powers of natural numbers, extending Nichomachus' observation (100 A.D) that the sum of cubes is the square of the sum of linear powers. Faulhaber's observations were proven by C. Jacobi two hundred years later. 

We use a recent insight of J. Conway to use the formal Integral to shed new light on Faulhaber's polynomials and to extend his insights. And we point out a mystery in the cases of 9th and 10th powers.