  We introduce the notion of finite multiple harmonic q-series at a primitive
root of unity and show that these specialize to the finite multiple zeta value
(FMZV) and the symmetrized multiple zeta value (SMZV) through an algebraic and
analytic operation, respectively. Further, we obtain families of linear
relations among these series which induce linear relations among FMZVs and
SMZVs of the same form. This gives evidence towards a conjecture of Kaneko and
Zagier relating FMZVs and SMZVs. Motivated by the above results, we define
cyclotomic analogues of FMZVs, which conjecturally generate a vector space of
the same dimension as that spanned by the finite multiple harmonic q-series at
a primitive root of unity of sufficiently large degree.
