On Multiple Zeta Values of Even Arguments

Abstract

For k <= n, let E(2n,k) be the sum of all multiple zeta values with even arguments whose weight is 2n and whose depth is k. Of course E(2n,1) is the value of the Riemann zeta function at 2n, and it is well known that E(2n,2) = (3/4)E(2n,1). Recently Z. Shen and T. Cai gave formulas for E(2n,3) and E(2n,4). We give two formulas form E(2n,k), both valid for arbitrary k <=n, one of which generalizes the Shen-Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers E(2n,k) and for the analogous numbers E*(2n,k) defined using multiple zeta-star values of even arguments.

DESY number added; misprints fixed; reference added. Second revision (2016): New result on multiple zeta-star values added

References in corpus (2)

• Extended double shuffle relations and the generating function of triple zeta values of any fixed weight
• A study on multiple zeta values from the viewpoint of zeta-functions of root systems

Cited by in corpus (13)

• Quasi-shuffle products revisited
• Generalized Harmonic Number Sums and Quasi-Symmetric Functions
• Duality of Weighted Sum Formulas of Alternating Multiple $T$-Values
• Some relations deduced from regularized double shuffle relations of multiple zeta values
• Symmetry results for multiple $t$-values
• Use of the generating function to generalize the sum formula for quadruple zeta values
• Sum formulas of mltiple zeta values with arguments are multiple of a positive integer
• A parameterized generalization of the sum formula for quadruple zeta values
• Structural identities for generalized multiple zeta values
• Weighted sum formulas of multiple zeta values with even arguments
• Infinite products involving Dirichlet characters and cyclotomic polynomials
• Restricted Sum Formula of Multiple Zeta Values
• Restricted Sum Formula of Alternating Euler Sums