From evaluating products of Wigner 6-j symbols I end up with expressions containing fractions of Gamma functions. For example,
$\sqrt{\frac{\Gamma (2 N)}{\Gamma (2 N+3)}},$
where $N$ is a positive integer. This expression can be simplified to
$\frac{1}{2 \sqrt{N (N+1) (2 N+1)}}.$
However, when executing
FullSimplify[Sqrt[Gamma[2*N]/Gamma[3 + 2*N]],{N >= 1, N \[Element] Integers}]
the unaltered expression containing the Gamma functions is returned. Funny enough, running FullSimplify on the inverse, that is,
FullSimplify[(Sqrt[Gamma[2*N]/Gamma[3 + 2*N]])^(-1),{N >= 1, N \[Element] Integers}]
does return
2*Sqrt[NN*(1 + NN)*(1 + 2*NN)]
However, as my expressions are more complicated than the example given here, simplifying the reciprocal is not an option for me. How can I make mathematica simplify this fractions of factorials?
P.S. I tried the rule suggested in this answer, but had no success.
FullSimplify[ Sqrt[Gamma[2*n]/Gamma[3 + 2*n]] // FunctionExpand, {n >= 1, n ∈ Integers}]
$\endgroup$N
for your own symbol by the way. $\endgroup$