I am interested in the quantity
$$
a_{n} = \sqrt{n/2} \frac{\Gamma((n-1)/2)}{\Gamma(n/2)}$$
(this is the geometric bias of the non-central t-distribution with $n$ d.f.) After some plotting, my hunch is that
$a_n \approx 1 + \frac{3}{4n} + \mathcal{O}\left(n^{-2}\right)$, as $n\to\infty$. Is there a known asymptotic result of this form? One may assume $n$ is an integer.
A little googling lead me to Feng Qi's excellent survey of inequalities around ratios of $\Gamma$ functions of this form, but I cannot find an asymptotic expansion of this form.
Best Answer
You're right that $a_n = 1 + \frac{3}{4n} + O(n^{-2})$. A reference is NIST's Digital Library of Mathematical Functions, Equation 5.11.12, where they have $$\frac{\Gamma(z+a)}{\Gamma(z+b)} \sim z^{a-b} \sum_{k=0}^{\infty} \frac{G_k(a,b)}{z^k},$$ where $G_k(a,b) = \binom{a-b}{k} B^{(a-b+1)}_k(a),$ and $B_n^{(i)}(x)$ is a generalized Bernoulli polynomial.
Your question has $z = n/2$, $a = -1/2$, $b = 0$.
If you take the DLMF formula out one more term, and using $G_0(a,b) =1$, $G_1(a,b) = \frac{1}{2} (a-b)(a+b-1)$, and $G_2(a,b) = \frac{1}{12} \binom{a-b}{2} (3(a+b-1)^2-(a-b+1))$, you get $$a_n = 1 + \frac{3}{4n} + \frac{25}{32n^2} + O(n^{-3}).$$
Added: Believe it or not, Wolfram Alpha can do this for you as well. For example, the link says that $$a_n = 1 + \frac{3}{4n} + \frac{25}{32n^2} + \frac{105}{128n^3} + \frac{1659}{2048 n^4} + \frac{6237}{8192 n^5} + O(n^{-11/2}).$$