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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).

MSC:

11B57 Farey sequences; the sequences \(1^k, 2^k, \dots\)
11M06 \(\zeta (s)\) and \(L(s, \chi)\)

Citations:

Zbl 0797.11026
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Full Text: DOI

References:

[1] Conway, John; Guy, Richard, The book of numbers, (1996), 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 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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