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Explicit evaluations of extended Euler sums. (English) Zbl 1097.11044

Let \(p, q >1\) and \(k\) be positive integers. In the paper two new extended Euler sums \[ E_{p,q}^{(k)}=\sum_{n=1}^{\infty}\frac{1}{n^q}\sum_{r=1}^{kn}\frac{1}{r^p} \quad \text{and} \quad T_{p,q}^{(k)}=\sum_{n=1}^{\infty}\frac{1}{n^q}\sum_{r=1}^{\frac{n}{k}}\frac{1}{r^p} \] are defined and their expressions in terms of values of \(\operatorname{Re} E_{s}(x)\), \(\operatorname{Im} E_s(x)\), and of values of the Riemann zeta-function \(\zeta(s)\) at positive integers, for \(p+q\) odd, and \(p>1\), \(q>1\), are presented. Here \(E_s(x)\), \(x \in \mathbb{R}\), is the periodic zeta-function, for \(\operatorname{Re} s >1\), defined by \[ E_s(x) =\sum_{n=1}^{\infty}\frac{e^{2 \pi i n x}}{n^s}, \] and \(\operatorname{Re} E_0(x)=-\frac{1}{2}\) for \(x>0\). The case of \(E_{1,2n}^{(k)}\) and \(T_{1,2n}^{(k)}\) is considered separately. For example, the authors prove that \[ E_{1,2}^{(2)}=\frac{11}{4}\zeta(3), \quad T_{1,2n}^{(2)}=\frac{9}{4}\zeta(3)-\frac{3}{2}\log 2 \cdot \zeta(2). \] General results are too complicated to be stated here. They generalize formulas for the classical Euler sum \[ \sum_{n=1}^{\infty}\frac{1}{n^q}\sum_{r=1}^{n}\frac{1}{r^p} \] presented by [B. C. Berndt, Ramanujan’s notebook. Part I. New York etc.: Springer (1985; Zbl 0555.10001)], D. Borwein, J. M. Borwein, R. Girgensohn [Proc. Edinb. Math. Soc. (2) 38, No. 2, 277–294 (1995; Zbl 0819.40003)], R. E. Crandall, J. P. Buhler [Exp. Math. 3, No. 4, 275–285 (1994; Zbl 0833.11045)], P. Flajolet, B. Salvy [Exp. Math. 7, No. 2, 15–35 (1998; Zbl 0920.11061)] and R. Sitaramachandrarao [J. Number Theory 25, No. 1, 1–19 (1987; Zbl 0606.10032)].

MSC:

11M35 Hurwitz and Lerch zeta functions
11B68 Bernoulli and Euler numbers and polynomials
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
40B05 Multiple sequences and series
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