On a lifting of $t$-adic symmetric multiple zeta values

The $t$-adic symmetric multiple zeta value is a generalization of the symmetric multiple zeta value from the perspective of the Kaneko-Zagier conjecture. In this paper, we introduce a further generalization with a new parameter $s$, which we call the $(s,t)$-adic symmetric multiple zeta value. Then, the $(s,t)$-adic version of the $t$-adic double shuffle relations, duality and cyclic sum formula are established. A finite counterpart of the $(s,t)$-adic symmetric multiple zeta value is also discussed.