New Results for Euler Sums

Abstract

We present a large number of analytic evaluations of Euler sums, namely sums such as \begin{align} M(m,n_0,n_1,n_2, \ldots, n_t) &= \sum_{k=1}^\infty \frac{H(k)^m}{k^{n_0} (k+1)^{n_1} (k+2)^{n_2} \cdots (k+t)^{n_t}}, \nonumber \end{align} for nonnegative integers $m$ and $(n_i)$, with $m \geq 1$ and $n_0 + n_1 + \cdots + n_t \geq 2$, where $H(k) = \sum_{j=1}^k 1/j$ is the harmonic function. These results were obtained either by algebraic manipulations, or else by very high-precision numerical evaluations combined with an integer relation algorithm to obtain the analytic formulas. We show how many of these results can be derived from a few basic facts, and that these techniques are applicable to Euler sums of even more general forms than the above cases. We then show that these results permit the calculation of constants for Euler sums resembling the Stieltjes $\gamma$ constants arising in the theory of the Riemann zeta function, and we also present some preliminary results on the asymptotic behavior of these constants.