The generalization of Faulhaber’s formula to sums of non-integral powers. (English) Zbl 1176.11004
Authors’ summary: “A formula for the sum of any positive-integral power of the first \(N\) positive integers was published by Johann Faulhaber (1580–1635). In this paper, we generalize Faulhaber’s formula to non-integral complex powers with real part greater than \(-1\).”
See also the following article of Donald Knuth, Johann Faulhaber and sums of powers. Math. Comput. 61, No. 203, 277–294 (1993; Zbl 0797.11026).
See also the following article of Donald Knuth, Johann Faulhaber and sums of powers. Math. Comput. 61, No. 203, 277–294 (1993; Zbl 0797.11026).
Reviewer: Olaf Ninnemann (Berlin)
MSC:
| 11B57 | Farey sequences; the sequences \(1^k, 2^k, \dots\) |
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
Keywords:
sums of non-integral powers; Faulhaber’s formula; Bernoulli numbers; Euler–MacLaurin summation; Riemann zeta functionCitations:
Zbl 0797.11026References:
| [1] | Conway, John; Guy, Richard, The Book of Numbers (1996), Springer-Verlag: Springer-Verlag New York, p. 106 · Zbl 0866.00001 |
| [2] | Ivić, Aleksandar, The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function with Applications (1985), John Wiley & Sons: John Wiley & Sons New York · Zbl 0556.10026 |
| [3] | Parks, Harold R., Sums of non-integral powers, J. Math. Anal. Appl., 297, 343-349 (2004) · Zbl 1160.11310 |
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