The rank $n$ Weyl $A_n(\mathsf{k})$ algebra over a field $\mathsf{k}$ of zero characteristic does not satisfies any polinomial identity. If it were a PI-algebra, Kaplansky theorem would apply (since the Weyl algebra in this case is simple and hence primitive), and we would have that $A_n(k)$ is finite dimensional over its center; which is false.

The situation changes dramatically in case $char \, \mathsf{k}=p$. In this case $A_n(\mathsf{k})$ is a finite algebra over its center, which is $k[x_1^p,\ldots,x_n^,\partial_1^p,\ldots,\partial_n^p]$, generated by $N=2n(p-1)$ elements. Hence it satisy the standard identity of degree M, for every $M \geq N$.

What else is known about the $T$-ideal of polynomial identities of $A_n(\mathsf{k})$ when $char \, \mathsf{k}=p>0$?