Is there a way to simplify this quotient of factorials?

Question

So I'm implementing numerically this quotient of factorials (in spherical harmonics) and after $\nu=170$ I'm getting an overload error and I was wondering if there was a way to simplify this expression:

$$\frac{\Gamma{(\nu + \mu + 1)}}{\Gamma{(\nu - \mu + 1})}$$

for $0 \leq \mu \leq \nu$ and the gamma function is defined by: $\Gamma(n) = (n-1)!$.

Answer 1

For $\mu,\nu \in \mathbb{Z}$ and $0 < \mu \leq \nu$, $$ \frac{(\nu+\mu)!}{(\nu-\mu)!} = (\nu+\mu)(\nu+\mu-1)\dotsb (\nu-\mu+1), $$ by cancelling repeated terms. (This actually only needs $\mu$ to be an integer or half-integer: the product contains $2\mu$ terms.)