Evaluation of Triple Euler Sums

  • Jonathan M. Borwein
  • Roland Girgensohn


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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