A generating function identity for finite and symmetric multiple zeta values via Dougall's bilateral ${}_2{H}_2$ summation

We establish a generating function identity for finite and symmetric multiple zeta values using Dougall’s bilateral ${}_2{H}_2$ summation formula. As an application, we prove a restricted sum formula in which the parity of each component of the index is prescribed. We also derive a generating function identity for finite multiple zeta values ${\zeta }_{\mathcal{A}}^{}(\{1,2{\}}^n)$ and symmetric multiple zeta values ${\zeta }_{\mathcal{S}}^{}(\{1,2{\}}^n)$, thereby proving Kaneko’s conjecture for finite multiple zeta values and its symmetric analogue.