Representation of some special functions on transcendence basis

Abstract

The special functions such as multiple harmonic sums, polyzetas or multiple polylogarithm functions are compatible with quasi-shuffle algebras. By using transcendence bases of the quasi-shuffle algebras studied in the paper [4], we will express non-commutative generating series of these special functions and then identify on the local coordinates to reduce their polynomial relations or asymptotic expansions indexed by these bases.