  In this paper, we study various twisted A-harmonic sums, named following the
seminal log-algebraicity papers of G. Anderson. These objects are partial sums
of new types of special zeta values introduced by the first author and linked
to certain rank one Drinfeld modules over Tate algebras in positive
characteristic by Angl\`es, Tavares Ribeiro and the first author. We prove, by
using techniques introduced by the second author, that various infinite
families of such sums may be interpolated by polynomials, and we deduce, among
several other results, properties of analogues of finite zeta values but inside
the framework of the Carlitz module. In the theory of finite multi-zeta values
in characteristic zero, finite zeta values are all zero. In the Carlitzian
setting, there exist non-vanishing finite zeta values, and we study some of
their properties in the present paper.
