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On generalized harmonic numbers, Tornheim double series and linear Euler sums
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Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a linear combination of Tornheim double series of the same weight. New closed form evaluations of various Euler sums are presented. Finally certain combinations of linear Euler sums that are reducible to Riemann zeta values are discovered.
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On an infinite number of nonlinear Euler sums
Eight families of quadratic Euler sums of odd order are reduced to zeta values and polylogarithms, but order-7 reductions remain incomplete for two unresolved nonlinear sums.
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