  In 1935 Carlitz introduced Bernoulli-Carlitz numbers as analogues of
Bernoulli numbers for the rational function field $\mathbb F_r(T)$. In this
paper, we introduce Cauchy-Carlitz numbers as analogues of Cauchy numbers. By
using Stirling-Carlitz numbers, we give their arithmetical and combinatorial
properties and relations with Bernoulli-Carlitz numbers for $\mathbb F_r(T)$.
Several new identities are also obtained by using Hasse-Teichim\"uller
derivatives.
