  In this paper, we introduce the concept of the (higher order) Appell-Carlitz
numbers which unifies the definitions of several special numbers in positive
characteristic, such as the Bernoulli-Carlitz numbers and the Cauchy-Carlitz
numbers.Their generating function is usually named Hurwitz series in the
function field arithmetic. By using Hasse-Teichm\"uller derivatives, we also
obtain several properties of the (higher order) Appell-Carlitz numbers,
including a recurrence formula, two closed forms expressions, and a determinant
expression.
  The recurrence formula implies Carlitz's recurrence formula for
Bernoulli-Carlitz numbers. Two closed from expressions implies the
corresponding results for Bernoulli-Carlitz and Cauchy-Carlitz numbers . The
determinant expression implies the corresponding results for Bernoulli-Carlitz
and Cauchy-Carlitz numbers, which are analogues of the classical determinant
expressions of Bernoulli and Cauchy numbers stated in an article by Glaisher in
1875.
