Approximation of a ratio of factorials When working with a Gamma function I ended up getting the following relation. I am wondering how to find a good (something better than Stirling's formula) asymptotic approximation of it
$$
\frac{(N/2+1)!}{(N/2+1/2)!}.
$$
and how to find C, such that 
$$
\frac{(N/2+1)!}{(N/2+1/2)!}\leq C
$$
 A: Let
$f(n)
=\frac{(n/2+1)!}{(n/2+1/2)!}
$.
I had come up with
if $n$ is even,
$f(n)
\approx \sqrt{n/2+1}
$
and
if $n$ is odd,
then
$f(n)
\approx \sqrt{n/2+3/2}
$
and then did a search.
This is
$f(n)
=\dfrac{\Gamma(n/2+2)}{\Gamma(n/2+3/2)}
$.
According to
the result quoted below,
this,
with $a=2, b=3/2$,
is,
since
$a-b = 1/2$,
$f(n)
= \sqrt{\dfrac{n}{2}}
\left(1+\dfrac{(1/2)(-1/2)}{2(n/2)}
+O\left(\dfrac1{n^2}\right) \right)
= \sqrt{\dfrac{n}{2}}
\left(1-\dfrac{1}{4n}
+O\left(\dfrac1{n^2}\right) \right)
$.
A Google search for
"Ratios of Gamma functions"
comes up with this
(among others):
https://msp.org/pjm/1951/1-1/pjm-v1-n1-p14-s.pdf
"THE ASYMPTOTIC EXPANSION OF A RATIO
OF GAMMA FUNCTIONS"
by
F. G.TRICOMI AND A.ERDELYI
Among the results:
As $z \to \infty$,
$\dfrac{\Gamma(z+a)}{\Gamma(z+b)}
=z^{a-b}\left(1+\dfrac{(a-b)(a-b-1)}{2z}+O\left(|z|^{-2}\right)\right)
$.
An asymptotic expansion
$\dfrac{\Gamma(z+a)}{\Gamma(z)}
=z^a\sum_{n=0}^{\infty}A_n(a)z^{-n}
$
where
$A_0(a) = 1,
A_1(a) = \binom{a}{2},
A_2(a) = \frac{3a-1}{4}\binom{a}{3},
A_3(a) = \binom{a}{2}\binom{a}{4}
$
and,
in general,
$A_n(a)
=\dfrac1{n}\sum_{m=0}^{n-1}\binom{a-m}{n-m-1}A_m(a)
$.
$\dfrac{\Gamma(z+a)}{\Gamma(z+b)}
\sim \sum_{n=0}^{\infty} C_n(a-b, b)z^{a-b-n}
$
where
$C_0(a-b, b)
=1,
C_1(a-b, b)
=\dfrac12(a-b)(a-b-1),\\
C_2(a-b, b)
=\dfrac1{12}\binom{a-b}{2}(3(a+b-1)^2-a+b-1),\\
$
and,
in general,
$C_n(a-b, b)
=\dfrac1{n}\sum_{m=0}^{n-1}\left[\binom{a-b-m}{n-m+1}-(-1)^{n+m}(a-b)b^{n-m} \right]C_m(a-b, b)
$.
