Appell-Carlitz numbers

Abstract

In this paper, we introduce the concept of the (higher order) Appell-Carlitz numbers which unifies the denfiitions of several special numbers in positive characteristic, such as the Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. Their generating function is named Hurwitz series in the function field arithmetic ([11, p. 352, Definition 9.1.4]). By using Hasse-Teichmuller 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.