Is the hypergeometric function $_2F_1\left[\frac{1}{2},n+\frac{1}{2};n+1;z\right]$ expressible in terms of more elementary functions?

Question

Is this special case of hypergeometric function expressible in terms of more elementary functions :

$$_2F_1\left[\frac{1}{2},n+\frac{1}{2};n+1;z\right]$$

It will also be helpful for me to know, if this function appears in any well studied discrete probability functions.

$\textbf{Note :}$ $n \in \mathbb{Z}_{}^{+}$ and $z \in \mathbb{R}$ with $0\leq z \leq 1$.

Answer 1

Maple relates this to the associated Legendre polynomials/functions: $$_2F_1\left(\frac{1}{2},n+\frac{1}{2};n+1;z\right) = {z}^{-n/2}P^{-n}_{-1/2}\left(-{\frac {z+1}{z-1}}\right) \frac{\Gamma(n+1)}{\sqrt{1-z}}$$

See also Abramowitz/Stegun 15.4.15 with $a=1/2, b=n+1/2$