Differential Calculus on $\mathbf{h}$-Deformed Spaces

We construct the rings of generalized differential operators on the $\mathbf{h}$-deformed vector space of $\mathbf{gl}$-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 $\mathbf{h}$-deformed differential operators $\operatorname{Diff}_{\mathbf{h},\sigma}(n)$ 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 $\operatorname{Diff}_{\mathbf{h},\sigma}(n)$.