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Abstract for 
Jonathan M. Borwein and Roland Girgensohn, Evaluation of Triple Euler Sums

Let $a,b,c$ be positive integers and define the so-called triple, double and
single Euler sums by
$$\zeta(a,b,c) \ := \ 
\sum_{x=1}^{\infty} \sum_{y=1}^{x-1} \sum_{z=1}^{y-1} {1 \over x^a y^b z^c},$$
$$
\zeta(a,b) \ := \ \sum_{x=1}^\infty \sum_{y=1}^{x-1} {1 \over x^a y^b} \quad
$$
and
$$
\zeta(a) \ := \ \sum_{x=1}^\infty {1 \over x^a}.$$
Extending earlier work about double sums,
we prove that whenever $a+b+c$ is even or less than~10, then $\zeta(a,b,c)$
can be expressed as a rational linear combination of products of double and
single Euler sums. The proof involves finding and solving linear equations
which relate the different types of sums to each other. We also sketch some
applications of these results in Theoretical Physics.


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