# Truncated Bernoulli-Carlitz and truncated Cauchy-Carlitz numbers

## TL;DR

This paper introduces truncated Bernoulli-Carlitz and Cauchy-Carlitz numbers as new analogues and extensions of existing special numbers, expressed explicitly via incomplete Stirling-Carlitz numbers.

## Contribution

It defines and explores properties of these new truncated numbers, extending the theory of Bernoulli and Cauchy numbers in the Carlitz setting.

## Key findings

- Explicit formulas in terms of incomplete Stirling-Carlitz numbers
- Extensions of classical Bernoulli and Cauchy numbers
- New analogues for hypergeometric Bernoulli and Cauchy numbers

## Abstract

In this paper, we define the truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers as analogues of hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, and as extensions of Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of incomplete Stirling-Carlitz numbers.