  Siegel-Shidlovskii theory of $E$-functions involves a non-vanishing proof for
the determinants attached to the linear forms $D^kR(t)$, derivatives of an
auxiliary function $R(t)$. Let a non-zero function $F(t)$ satisfy $m$th order
linear differential equation which we shall write using the differential
operator $\Delta=tD$ and let $L(t)$ be any non-zero linear form of the
derivatives $\Delta^i F(t)$ $(i=0,...,m-1; m\ge 2)$. The determinants
$\det\mathcal A_k$ attached to the linear forms $\Delta^kL(t)$ have certain
simple properties that allow us to give a short proof for the non-vanishing of
$\det\mathcal A_k$ for a class of differential equations including a subclass
of hypergeometric differential equations.
