# 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?

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 $$0\leq z \leq 1$$.

• I guess $n$ is an integer? Positive? – Fabian Dec 15 '18 at 21:00
• @Fabian Yes, I forgot to mention. – Sunyam Dec 15 '18 at 21:01

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 $$a=1/2, b=n+1/2$$
• Thank you. Mathematica was not able to simplify. Does this hold for complex $z$. – Sunyam Dec 15 '18 at 21:52
• Yes, for complex $z$ (with some restrictions) this is 15.4.14 on the same A/S page. – gammatester Dec 15 '18 at 22:09
• The first few of these (and I assume all of them) have the form $$\frac{A(z)}{\pi}\;K(\sqrt{z})+\frac{B(z)}{\pi}\;F(\sqrt{z})$$where $A(z), B(z)$ are rational functions and $F,K$ are the complete elliptic integrals. (I used Maple, perhaps as usual the $\sqrt{z}$ would be $z$ in Mathematica notation.) – GEdgar Dec 15 '18 at 22:33