Weighted sum formula for multiple zeta values

Weighted Sum Formula for Multiple Zeta Values

Abstract. The sum formula is a basic identity of multiple zeta values that expresses a Riemann zeta value as a homogeneous sum of multiple zeta values of a given dimension. This formula was already known to Euler in the dimension two case, conjectured in the early 1990s for higher dimensions and then proved by Granville and Zagier independently. Recently a weighted form of Euler’s formula was o...

ON THE SUM FORMULA FOR MULTIPLE q-ZETA VALUES

Abstract. Multiple q-zeta values are a 1-parameter generalization (in fact, a q-analog) of the multiple harmonic sums commonly referred to as multiple zeta values. These latter are obtained from the multiple q-zeta values in the limit as q → 1. Here, we discuss the sum formula for multiple q-zeta values, and provide a self-contained proof. As a consequence, we also derive a q-analog of Euler’s ...

New Families of Weighted Sum Formulas for Multiple Zeta Values

SUM OF MULTIPLE q-ZETA VALUES

The generating function of the sums of multiple q-zeta values with fixed weights, depths and 1-heights, 2-heights, . . . , r-heights is represented in terms of specializations of basic hypergeometric functions.

Some Notes on Weighted Sum Formulae for Double Zeta Values

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 fr...