Cauchy-Carlitz numbers

TL;DR

This paper introduces Cauchy-Carlitz numbers as analogues of classical Cauchy numbers within the function field setting, exploring their properties, relations, and identities using Stirling-Carlitz numbers and Hasse-Teichmüller derivatives.

Contribution

It is the first to define and analyze Cauchy-Carlitz numbers, establishing their properties and connections with Bernoulli-Carlitz numbers in the context of rational function fields.

Findings

Derived arithmetical and combinatorial properties of Cauchy-Carlitz numbers.

Established relations between Cauchy-Carlitz and Bernoulli-Carlitz numbers.

Obtained new identities using Hasse-Teichmüller derivatives.

Abstract

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.