On Andr\'e periods of mixed Tate motives

In this note, we show that the $p$-adic periods of motives introduced recently by Ancona and Fr\u{a}\c{t}il\u{a} (``Andr\'e periods'') reduce to the classically studied notion in the case of Mixed Tate motives. We also connect Andr\'e periods with Coleman integration by observing that the Frobenius-fixed de Rham paths of Besser and Vologodsky come from motivic paths in characteristic $p$ (unconditionally in the mixed Tate setting, conditionally in general). We use this to realize special values of $p$-adic multiple polylogarithms as Andr\'e periods in a concrete way.