On the Evaluation of Euler Sums
- 1 January 1994
- journal article
- research article
- Published by Taylor & Francis in Experimental Mathematics
- Vol. 3 (4) , 275-285
- https://doi.org/10.1080/10586458.1994.10504297
Abstract
Euler studied double sums of the form 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 ζ(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.Keywords
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