  Anderson generating functions have received a growing attention in function
field arithmetic in the last years. Despite their introduction by Anderson in
the 80s where they were at the heart of comparison isomorphisms, further
important applications e.g. to transcendence theory have only been discovered
recently. The Anderson-Thakur special function interpolates L-values via
Pellarin-type identities, and its values at algebraic elements recover
Gauss-Thakur sums, as shown by Angl\`es and Pellarin. For Drinfeld-Hayes
modules, generalizations of Anderson generating functions have been introduced
by Green-Papanikolas and -- under the name of `special functions' -- by
Angl\`es-Ngo Dac-Tavares Ribeiro. In this article, we provide a general
construction of special functions attached to any Anderson A-module. We show
direct links of the space of special functions to the period lattice, and to
the Betti cohomology of the A-motive. We also undertake the study of
Gauss-Thakur sums for Anderson A-modules, and show that the result of
Angl\`es-Pellarin relating values of the special functions to Gauss-Thakur sums
holds in this generality.
