# A Note About the {Ki(z)} Functions

## TL;DR

This paper simplifies proofs of properties of the $K_{i}(z)$ functions, addresses an open question from prior work, and explores their differential transcendency, contributing to the theoretical understanding of these special functions.

## Contribution

It provides simplified proofs of known properties, resolves an open question, and discusses the differential transcendency of the $K_{i}(z)$ functions.

## Key findings

- Simplified proofs of two properties of $K_{i}(z)$ functions.
- Resolution of an open question from previous research.
- Discussion on the differential transcendency of $K_{i}(z)$ functions.

## Abstract

In the article [Petojevic 2006], A. Petojevi\' c verified useful properties of the $K_{i}(z)$ functions which generalize Kurepa's [Kurepa 1971] left factorial function. In this note, we present simplified proofs of two of these results and we answer the open question stated in [Petojevic 2006]. Finally, we discuss the differential transcendency of the $K_{i}(z)$ functions.