Simplification of Gamma functions

Question

I have the following function involving Gamma functions where $d,t\in \mathbb{Z}$ and $d,t >0$ such that $$f(d,t) = \frac{\displaystyle\Gamma\left(\frac{d-1}{t}\right)\cdot \Gamma\left(\frac{d}{2}\right)}{\displaystyle\Gamma\left(\frac{d-1}{2}\right)\cdot \Gamma\left(\frac{d}{t}\right)}.$$ Can this expression be simplified further? Or Can anything useful/trends w.r.t. $d,t$ be extracted out of this function? Or can we give an approximate for this function?

Answer 1

I do not think that you could simplify the expression.

However, as @Greg Martin commented, if $d$ is large, take logarithms, use four times Stirling approxiamation and continue with Taylor series to get $$\log(f)=\frac{(t-2) \log (d)-t \log (2)+2 \log (t)}{2 t}-\frac{t-2}{4 d t}+\frac{(t-2) (t-1)}{12 d^2 t}+O\left(\frac{1}{d^3}\right)$$

Let us try for $t=10$ and $d=50$; the "exact" value is $\log(f)=1.44474$ while the above expansion gives $1.44473$.