Multiple zeta values and Euler sums

Abstract

In this paper, we establish some expressions of series involving harmonic numbers and Stirling numbers of the first kind in terms of multiple zeta values, and present some new relationships between multiple zeta values and multiple zeta star values. The relationships obtained allow us to find some nice closed form representations of nonlinear Euler sums through Riemann zeta values and linear sums. Furthermore, we show that the combined sums \[H\left( {a,b;m,p} \right) := \sum\limits_{a + b = m - 1} {ζ\left( {{\left\{ p \right\}_a},p + 1,{\left\{ p \right\}_b}} \right)}\quad (m\in \N,p>1) \] and \[{H^ \star }\left( {a,b;m,p} \right) := \sum\limits_{a + b = m - 1} {{ζ^ \star }\left( {{\left\{ p \right\}_a},p + 1,{\left\{ p \right\}_b}} \right)}\quad (m\in \N,p>1) \] are reducible to polynomials in zeta values, and give explicit recurrence formulas. Some interesting (known or new) consequences and illustrative examples are considered.

arXiv admin note: text overlap with arXiv:1609.04924

References in corpus (1)

• Duality and (q-)multiple zeta values

Cited by in corpus (13)

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• On harmonic numbers and nonlinear Euler sums
• Evaluations of Euler type sums of weight $\le$ 5
• Explicit formulas of Euler sums via multiple zeta values
• Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series
• Logarithmic integrals, zeta values, and tiered binomial coefficients
• Explicit Relations between Multiple Zeta Values and Related Variants
• On q-analogues of quadratic Euler sums