The Gamma function $\Gamma(n+1)$ is equivalent to the factorial function $n!$ for $n\in \mathbb{N}$. It's a generalisation of $n!$ for $n \in \mathbb{\mathbb{R}}$.
In order to build a Frobenius serie, I would like to write someting like $(n+\frac{1}{2})!!$ for $n$ natural, but I can't since in this case $n+\frac{1}{2}$ will never belong to $\mathbb{N}$.
My question is: Is there a function similar to the Gamma function but for $n!!$ ?
Thank you in advance !