Appell-Carlitz numbers

TL;DR

This paper introduces higher order Appell-Carlitz numbers in positive characteristic, unifying several special numbers and deriving their properties using Hasse-Teichmüller derivatives, including recurrence, closed forms, and determinant expressions.

Contribution

It defines higher order Appell-Carlitz numbers and establishes their properties, connecting them to classical and Carlitz-specific numbers in positive characteristic.

Findings

Derived recurrence formulas for Appell-Carlitz numbers

Obtained closed form expressions for these numbers

Established determinant expressions analogous to classical cases

Abstract

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.