In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if
f
∈
A
t
p
(
U
)
f\in A_t^p(\mathcal {U})
and
L
α
f
≡
0
\mathcal {L}^{\alpha } f\equiv 0
for some nonnegative multi-index
α
\alpha
, then
f
≡
0
f\equiv 0
, where
L
α
≔
(
L
1
)
α
1
⋯
(
L
n
)
α
n
\mathcal {L}^{\alpha }≔(\mathcal {L}_1)^{\alpha _1} \cdots (\mathcal {L}_n)^{\alpha _n}
with
L
j
=
∂
∂
z
j
+
2
i
z
¯
j
∂
∂
z
n
\mathcal {L}_j = \frac {\partial }{\partial z_j} + 2i \bar {z}_j \frac {\partial }{\partial z_n}
for
j
=
1
,
…
,
n
−
1
j=1,\ldots , n-1
and
L
n
=
∂
∂
z
n
\mathcal {L}_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.