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Johann Faulhaber and Sums of Powers

Donald E. Knuth
Mathematics of Computation
Vol. 61, No. 203, Special Issue Dedicated to Derrick Henry Lehmer (Jul., 1993), pp. 277-294
DOI: 10.2307/2152953
Stable URL: http://www.jstor.org/stable/2152953
Page Count: 18
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Johann Faulhaber and Sums of Powers
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Abstract

Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the r-fold summation of 1m, 2m, ..., nm is a polynomial in n(n + r) when m is a positive odd number. The present paper explores a computation-based approach by which Faulhaber may well have discovered such results, and solves a 360-year-old riddle that Faulhaber presented to his readers. It also shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers. Faulhaber's coefficients can moreover be generalized to noninteger exponents, obtaining asymptotic series for 1α + 2α + ... + nα in powers of n-1(n + 1)-1.

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