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1994 On the evaluation of Euler sums
Richard E. Crandall, Joe P. Buhler
Experiment. Math. 3(4): 275-285 (1994).

Abstract

Euler studied double sums of the form $$ \z(r,s)=\sum_{1\le m<n}{1\over n^sm^r} $$ for positive integers $r$ and $s$, and inferred, for the special cases $r = 1$ or $r+s$ odd, elegant identities involving values of the Riemann zeta function. Here we establish various series expansions of $\z(r, s)$ for real numbers $r$ and $s$. These expansions generally involve infinitely many zeta values. The series of one type terminate for integers $r$ and $s$ with $r+s$ odd, reducing in those cases to the Euler identities. Series of another type are rapidly convergent and therefore useful in numerical experiments.

Citation

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Richard E. Crandall. Joe P. Buhler. "On the evaluation of Euler sums." Experiment. Math. 3 (4) 275 - 285, 1994.

Information

Published: 1994
First available in Project Euclid: 24 March 2003

zbMATH: 0833.11045
MathSciNet: MR1341720

Subjects:
Primary: 11M41
Secondary: 11Y35

Keywords: incomplete gamma function , Lerch-Hurwitz zeta function , periodic zeta function , polylogarithms , Riemann zeta function

Rights: Copyright © 1994 A K Peters, Ltd.

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Vol.3 • No. 4 • 1994
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