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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


Johann Faulhaber and sums of powers

Author: Donald E. Knuth
Journal: Math. Comp. 61 (1993), 277-294
MSC: Primary 11B83; Secondary 01A45, 01A55, 11B57
MathSciNet review: 1197512
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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 $ {1^m},{2^m}, \ldots ,{n^m}$ 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 $ {1^\alpha } + {2^\alpha } + \cdots + {n^\alpha }$ in powers of $ {n^{ - 1}}{(n + 1)^{ - 1}}$.

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PII: S 0025-5718(1993)1197512-7
Article copyright: © Copyright 1993 American Mathematical Society