On variants of symmetric multiple zeta-star values and the cyclic sum formula

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The t-adic symmetric multiple zeta values were defined by Jarossay, which have been studied as a real analogue of the \({\varvec{p}}\)-adic finite multiple zeta values. In this paper, we consider the star analogues based on several regularization processes of multiple zeta-star values: harmonic regularization, shuffle regularization, and Kaneko–Yamamoto’s type regularization. We also present the cyclic sum formula for t-adic symmetric multiple zeta(-star) values, which is the counterpart of that for \({\varvec{p}}\)-adic finite multiple zeta(-star) values obtained by Kawasaki. The proof uses our new relationship that connects the cyclic sum formula for t-adic symmetric multiple zeta-star values and that for the multiple zeta-star values.