Ratios of gamma functions come up often in applications. If the two gamma function arguments differ by an integer, then it’s easy to calculate their ratio exactly by using (repeatedly if necessary) the fact at Γ(x + 1) = x Γ(x).
If the arguments differ by 1/2, there is no closed formula, but the there are useful approximations. I’ve needed something like this a few times lately.
The simplest approximation is
You could motivate or interpret this as saying Γ(x + 1/2) is approximately the geometric mean between Γ(x + 1) and Γ(x). As we’ll see in the plot below, this approximation is good to a couple significant figures for moderate values of x.
There is another approximation that is a little more complicated but much more accurate.
The following plot shows the relative error in both approximations.
By the way, the first approximation above is a special case of the more general approximation
Source: J. S. Frame. An Approximation to the Quotient of Gamma Function. The American Mathematical Monthly, Vol. 56, No. 8 (Oct., 1949), pp. 529-535