Some Notes on Weighted Sum Formulae for Double Zeta Values

TL;DR

This paper introduces elementary proofs for weighted sum formulas of double zeta values, presents new evaluations related to harmonic numbers and special zeta functions, and discusses methods for discovering similar sums.

Contribution

It provides a unified elementary approach to prove known formulas and derives new sum evaluations involving multiple zeta-related functions.

Findings

Elementary proofs of three weighted sum formulas for double zeta values

New evaluations involving harmonic numbers, alternating double zeta values, and Witten zeta function

Recursion-based sums involving Riemann zeta and Dirichlet beta functions

Abstract

We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta values, and the Witten zeta function. We discuss a heuristic for finding or dismissing the existence of similar simple sums. We also produce some new sums from recursions involving the Riemann zeta and the Dirichlet beta functions.