Linear natural liftings of forms to Weil bundles with Weil algebras ${\mathbb{D}}_k^r$

We give an explicit description and calculate the dimension of the vector space of linear natural liftings of $p$-forms on $n$-dimensional manifolds $M$ to $q$-forms on $T^{{\mathbb{D}}_k^r}M$, where ${\mathbb{D}}_k^r$ is the Weil algebra of $r$-jets at 0 of smooth functions ${\mathbb R}^k\longrightarrow{\mathbb R}$,forallnon-negativeintegers$n$,$p$,$q$,$r$,$k$exceptthecase$p=n$and$q=0$.