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A new method in the study of Euler sums. (English) Zbl 1155.40002
Summary: A new method in the study of Euler sums is developed. A host of Euler sums, typically of the form $\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}\sum_{m=1}^{n}\frac{g(m)}{m^{t}}$, are expressed in closed form. Also obtained as a by-product are some striking recursive identities involving several Dirichlet series including the well-known Riemann zeta function.

40A25Approximation to limiting values (summation of series, etc.)
40B05Multiple sequences and series
11M99Analytic theory of zeta and L-functions
33E99Other special functions
Full Text: DOI
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[2] Bailey, D.H., Borwein, J.M., Girgensohn, R.: Experimental evaluation of Euler sums. Exp. Math. 3, 17--30 (1994) · Zbl 0810.11076
[3] Basu, A., Apostol, Tom M.: A new method for investigating Euler sums. Ramanujan J. 4, 397--419 (2000) · Zbl 0971.40001 · doi:10.1023/A:1009868016412
[4] Crandall, R.E., Buhler, J.P.: On the evaluation Euler sums. Exp. Math. 3, 275--285 (1994) · Zbl 0833.11045
[5] Ramanujan, S.: Note Books, vol. 2 (1957) · Zbl 0077.06401
[6] Williams, G.T.: A method of evaluating {$\zeta$}(2n). Am. Math. Mon. 60, 19--25 (1953) · Zbl 0050.06803 · doi:10.2307/2306473