In this paper, we establish a new asymptotic expansion of a ratio of two gamma functions, that is, as
x
→
∞
x\rightarrow \infty
,
[
Γ
(
x
+
u
)
Γ
(
x
+
v
)
]
1
/
(
u
−
v
)
∼
(
x
+
σ
)
exp
[
∑
k
=
1
m
B
2
n
+
1
(
ρ
)
w
n
(
2
n
+
1
)
(
x
+
σ
)
−
2
k
+
R
m
(
x
;
u
,
v
)
]
,
\begin{equation*} \left [ \frac {\Gamma \left ( x+u\right ) }{\Gamma \left ( x+v\right ) }\right ] ^{1/\left ( u-v\right ) }\thicksim \left ( x+\sigma \right ) \exp \left [ \sum _{k=1}^{m}\frac {B_{2n+1}\left ( \rho \right ) }{wn\left ( 2n+1\right ) }\left ( x\!+\!\sigma \right ) ^{-2k}\!+\!R_{m}\left ( x;u,v\right ) \right ] , \end{equation*}
where
u
,
v
∈
R
u,v\in \mathbb {R}
with
w
=
u
−
v
≠
0
w=u-v\neq 0
and
ρ
=
(
1
−
w
)
/
2
\rho =\left ( 1-w\right ) /2
,
σ
=
(
u
+
v
−
1
)
/
2
\sigma =\left ( u+v-1\right ) /2
,
B
2
n
+
1
(
ρ
)
B_{2n+1}\left ( \rho \right )
are the Bernoulli polynomials. We also prove that the function
x
↦
(
−
1
)
m
R
m
(
x
;
u
,
v
)
x\mapsto \left ( -1\right ) ^{m}R_{m}\left ( x;u,v\right )
for
m
∈
N
m\in \mathbb {N}
is completely monotonic on
(
−
σ
,
∞
)
\left ( -\sigma ,\infty \right )
if
|
u
−
v
|
>
1
\left \vert u-v\right \vert >1
, which yields an explicit bound for
|
R
m
(
x
;
u
,
v
)
|
\left \vert R_{m}\left ( x;u,v\right ) \right \vert
and some new inequalities.