On harmonic numbers and nonlinear Euler sums

Abstract

In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic representations of Euler sums through values of polylogarithm function and Riemann zeta function. Moreover, we provide explicit evaluations for almost all Euler sums with weight 5, which can be expressed in terms of zeta values and polylogarithms. Furthermore, we give explicit formula for several classes of Euler-related sums in terms of zeta values and harmonic numbers, and several examples are given. The given representations are new.

References in corpus (6)

• Harmonic Sums and Mellin Transforms up to two-loop Order
• The Multiple Zeta Value Data Mine
• Multiple zeta values and Euler sums
• On a three-dimensional symmetric Ising tetrahedron, and contributions to the theory of the dilogarithm and Clausen functions
• Evaluations of nonlinear Euler sums of weight ten
• Some evaluation of cubic Euler sums

Cited by in corpus (3)

• On logarithmic integrals, harmonic sums and variations
• Evaluations of Euler type sums of weight $\le$ 5
• Some evaluation of cubic Euler sums