Consecutive evaluation of Euler sums
The Euler sums are \(S(r,p)=\sum_{n=1}^{\infty}{H_n^{(r)}}\), where \[ H_n^{(r)}={1\over 1^r}+{1\over 2^r}+\cdots+{1\over n^r},\quad r\geq 1,\;p\geq 2. \] Euler evaluated several \(S(r,p)\) in terms of the Riemann zeta function, and the numbers \(S(r,p)\) for \(p+r\) odd were first explicitly evaluated by \textit{D. Borwein, J.~M. Borwein} and \textit{R. Girgensohn} [Explicit evaluation of Euler sums, Proc. Edinb. Math. Soc., II. Ser. 38, 277-294 (1995; Zbl 0819.40003)]. In the present paper the author gives an elementary procedure for consecutive evaluation of \(S(r,p)\) for \(r=1,2,3,\ldots\) by a recursive relation. In addition to recalculating the values for \(r+p\) odd, certain \(S(r,p)\) with \(r+p\) even are obtained explicitly in terms of zeta values; others are obtained only in terms of zeta functions and \(S(k,l)\) where \(k+l=r+p\).
scientific article; zbMATH DE number 1780374