Evaluation of triple Euler sums

Jonathan M. Borwein ; Roland Girgensohn

The electronic journal of combinatorics, Tome 3 (1996) no. 1

Résumé

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.

Détail

DOI : 10.37236/1247

Classification : 11M32, 11Y50, 33E99, 40A25, 40B05
Mots-clés : Riemann zeta function, harmonic numbers, quantum field, knot theory, polylogarithms, Euler sums, double sums