Explicit formulas of the relation between multiple zeta functions of Arakawa–Kaneko and Euler–Zagier types

Multiple zeta functions of Arakawa–Kaneko and Euler–Zagier types are known as generalizations of the Riemann zeta function. In 2018, Kaneko and Tsumura proved that the multiple zeta functions of Arakawa–Kaneko type can be expressed as a [Formula: see text]-linear combination of products of the ones of Euler–Zagier type and multiple zeta values. In this paper, we give explicit formulas for the above-mentioned relation. Moreover, as a key ingredient of its proof, we also give certain functional equations among multi-polylogarithm functions explicitly.