Multiple series connected to Hoffman's conjecture on multiple zeta values

Recent results of Zlobin and Cresson–Fischler–Rivoal allow one to decompose any suitable p-uple series of hypergeometric type into a linear combination (over the rationals) of multiple zeta values of depth at most p; in some cases, only the multiple zeta values with 2's and 3's are involved (as in Hoffman's conjecture). In this text, we study the depth p part of this linear combination, namely the contribution of the multiple zeta values of depth exactly p. We prove that it satisfies some symmetry property as soon as the p-uple series does, and make some conjectures on the depth p−1 part of the linear combination when p=3. Our result generalizes the property that (very) well-poised univariate hypergeometric series involve only zeta values of a given parity, which is crucial in the proof by Rivoal and Ball–Rivoal that ζ(2n+1) is irrational for infinitely many n⩾1. The main feature of the proof is an algebraic approach, based on representations of p(Z/2Z)⋊Sp.