Taylor coefficients of Anderson-Thakur series and explicit formulae

For each positive characteristic multiple zeta value (defined by Thakur in Function field arithmetic, World Scientific Publishing, River Edge, 2004), the first and third authors in Chang and Mishiba (Invent Math, 2020, https://doi.org/10.1007/s00222020-00988-1) constructed a t-module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value. The objective of this paper is to use the Taylor coefficients of Anderson-Thakur series and t-motivic Carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector.