Bernoulli-Carlitz and Cauchy-Carlitz numbers with Stirling-Carlitz numbers

TL;DR

This paper explores the properties and formulas of Bernoulli-Carlitz and Cauchy-Carlitz numbers using Stirling-Carlitz numbers, extending their definitions and applications in function fields and hypergeometric contexts.

Contribution

It introduces the second analogue of Stirling-Carlitz numbers and derives new formulas for Bernoulli and Cauchy numbers in formal power series and function fields.

Findings

Derived explicit formulas for Bernoulli-Carlitz and Cauchy-Carlitz numbers.

Established connections between Stirling-Carlitz numbers and hypergeometric Bernoulli and Cauchy numbers.

Applied Hasse-Teichmüller derivatives to these numbers.

Abstract

Recently, the Cauchy-Carlitz number was defined as the counterpart of the Bernoulli-Carlitz number. Both numbers can be expressed explicitly in terms of so-called Stirling-Carlitz numbers. In this paper, we study the second analogue of Stirling-Carlitz numbers and give some general formulae, including Bernoulli and Cauchy numbers in formal power series with complex coefficients, and Bernoulli-Carlitz and Cauchy-Carlitz numbers in function fields. We also give some applications of Hasse-Teichm\"uller derivative to hypergeometric Bernoulli and Cauchy numbers in terms of associated Stirling numbers.