# Differential Calculus on h-Deformed Spaces

## TL;DR

This paper constructs and analyzes the rings of generalized differential operators on h-deformed vector spaces, revealing their dependence on rational functions and solving associated difference equations.

## Contribution

It introduces a family of h-deformed differential operator rings parameterized by rational functions, extending the understanding beyond the q-deformed case.

## Key findings

- The rings are labeled by rational functions satisfying difference equations.
- The general solution to the difference system is obtained.
- Properties of the rings are described in detail.

## Abstract

We construct the rings of generalized differential operators on the ${\bf h}$-deformed vector space of ${\bf gl}$-type. In contrast to the $q$-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of ${\bf h}$-deformed differential operators $\operatorname{Diff}_{{\bf h},\sigma}(n)$ is labeled by a rational function $\sigma$ in $n$ variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system and describe some properties of the rings $\operatorname{Diff}_{{\bf h},\sigma}(n)$.