2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71361We introduce and study new versions of polylogarithms and a zeta function on a completion of $\mathbb F_q (x)$ at a finite place. The construction is based on the use of the Carlitz differential equations for $\mathbb F_q$-linear functions.15 pages, LaTeX-2eNumber Theory11G09; 11S40; 11M38; 12H99text