Evaluation of Triple Euler Sums

1996/08/07 by Jonathan M. Borwein, Roland Girgensohn · 1 citation
Mathematics · Computer Science · #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics #semigroups and automata theory #Mathematics #Euler's formula #Combinatorics #Euler number (physics) #Discrete mathematics #Mathematical analysis #Euler equations #Backward Euler method

paper · pdf · doi:10.37236/1247

openalex publication_date 1996/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let a,b,c be positive integers and define the so-called triple, double and single Euler sums by ζ(a,b,c) := ∑_x=1^∞ ∑_y=1^x-1 ∑_z=1^y-1 1 \over x^a y^b z^c, ζ(a,b) := ∑_x=1^∞ ∑_y=1^x-1 1 \over x^a y^b and ζ(a) := ∑_x=1^∞ 1 \over x^a. Extending earlier work about double sums, we prove that whenever a+b+c is even or less than 10, then ζ(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.

Citations

• Transformation formulae for multiple series
• Euler Sums and Contour Integral Representations
• Explicit evaluation of Euler sums
• On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory
• Harmonic Double Series
• Renormalization and Knot Theory

Cited by

• Multiple zeta function [wikipedia]