  Let $\pi$ be an irreducible cuspidal representation of
$\mathrm{GL}_{kn}\left(\mathbb{F}_q\right)$. Assume that $\pi = \pi_{\theta}$,
corresponds to a regular character $\theta$ of $\mathbb{F}_{q^{kn}}^{*}$. We
consider the twisted Jacquet module of $\pi$ with respect to a non-degenerate
character of the unipotent radical corresponding to the partition $(n^k)$ of
$kn$. We show that, as a
$\mathrm{GL}_{n}\left(\mathbb{F}_q\right)$-representation, this Jacquet module
is isomorphic to $\pi_{\theta \upharpoonright_{\mathbb{F}_n^*}} \otimes
\mathrm{St}^{k-1}$, where $\mathrm{St}$ is the Steinberg representation of
$\mathrm{GL}_{n}\left(\mathbb{F}_q\right)$. This generalizes a theorem of D.
Prasad, who considered the case $k=2$. We prove and rely heavily on a
formidable identity involving $q$-hypergeometric series and linear algebra.
