FAULHABER,JOHANN AND SUMS OF POWERS

Early 17th-century mathematical publications of Johann Faulhaber conta in some remarkable theorems, such as the fact that the r-fold summatio n of 1m , 2m , ..., n(m) is a polynomial in n(n + r) when m is a posit ive odd number. The present paper explores a computation-based approac h by which Faulhaber may well have discovered such results, and solves a 360-year-old riddle that Faulhaber presented to his readers. It als o shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers. Faulhaber's coef ficients can moreover be generalized to noninteger exponents, obtainin g asymptotic series for 1alpha + 2alpha + ... + n(alpha) in powers of n-1(n + 1)-1.