  In this paper, we show that the Bergman functions on the Siegel upper
half-space enjoy the following uniqueness property: if $f\in A_t^p(\calU)$ and
$\bfL^{\alpha} f\equiv 0$ for some nonnegative multi-index $\alpha$, then
$f\equiv 0$, where $\bfL^{\alpha}:=(\bfL_1)^{\alpha_1} \cdots
(\bfL_n)^{\alpha_n}$ with $\bfL_j = \frac{\partial }{\partial z_j} + 2i
\bar{z}_j \frac{\partial }{\partial z_n}$ for $j=1,\ldots, n-1$ and $\bfL_n =
\frac{\partial }{\partial z_n}$. As a consequence, we obtain a new integral
representation for the Bergman functions on the Siegel upper half-space. In the
end, as an application, we derive a result that relates the Bergman norm to a
"derivative norm", which suggests an alternative definition of the Bloch space
and a notion of the Besov spaces over the Siegel upper half-space.
