In this paper, we study the asymptotic behavior of the smallest eigenvalue
λ
N
\lambda _N
, of the
(
N
+
1
)
×
(
N
+
1
)
(N+1)\times (N+1)
Hankel matrix
M
N
=
(
μ
j
+
k
)
0
≤
j
,
k
≤
N
\mathcal {M}_N=(\mu _{j+k})_{0\le j,k\le N}
generated by the semi-classical Hermite weight
w
(
z
,
t
)
=
|
z
|
λ
exp
(
−
z
2
+
t
z
)
,
z
,
t
∈
R
,
λ
>
−
1
w(z,t)=|z|^\lambda \exp \left (-z^2+tz\right ), z, t \in \mathbb {R}, \lambda >-1
. An asymptotic expression of the orthonormal polynomials
P
N
(
z
)
\mathcal {P}_N(z)
with the semi-classical Hermite weight
w
(
z
,
t
)
w(z,t)
is established as
N
N
tends to infinity. Based on the orthonormal polynomials
P
N
(
z
)
\mathcal {P}_N(z)
, we obtain the specific asymptotic formulas of
λ
N
\lambda _{N}
.