For $\alpha,\beta\in\mathbb{C}$ and $\gamma\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}$, Gauss' hypergeometric function ${}\_2F\_1(\alpha,\beta;\gamma;z)$ can be defined by the series
\begin{equation}\label{Gauss-HF-dfn}
{}\_2F\_1(\alpha,\beta;\gamma;z)=\sum\_{n=0}^{\infty}\frac{(\alpha)\_n(\beta)\_n}{(\gamma)\_n}\frac{z^n}{n!},\quad |z|<1.
\end{equation}
The following special cases are well-known:
\begin{align\*}
{}\_2F\_1(a,b;b;z)&=\frac{1}{(1-z)^a},\\
{}\_2F\_1(1,1;2;z)&=-\frac{\ln(1-z)}{z},\\
{}\_2F\_1\biggl(\frac12,1;\frac32;z^2\biggr)&=\frac1{2z}\ln\frac{1+z}{1-z},\\
{}\_2F\_1\biggl(\frac12,1;\frac32;-z^2\biggr)&=\frac{\arctan z}{z},\\
{}\_2F\_1\biggl(\frac12,\frac12;\frac32;z^2\biggr)&=\frac{\arcsin z}{z},\\
{}\_2F\_1\biggl(\frac12,\frac12;\frac32;-z^2\biggr)&=\frac{\ln\bigl(z+\sqrt{1+z^2}\bigr)}{z}.
\end{align\*}
See Chapter 5 and page 109 in the book [1] below.


Lemma 2.6 in the paper [2] below reads that, for $0\ne|t|<1$ and $n=1,2,\dotsc$,
\begin{equation\*}
{}\_2F\_1\biggl(\frac{1-n}{2}, \frac{2-n}{2};1-n;\frac1{t^2}\biggr)
=\frac{t}{2^n\sqrt{t^2-1}\,} \biggl[\biggl(1+\frac{\sqrt{t^2-1}\,}{t}\biggr)^n -\biggl(1-\frac{\sqrt{t^2-1}\,}{t}\biggr)^n\biggr].
\end{equation\*}


Corollary 4.1 in the paper [3] below states that, for $n=0,1,2,\dotsc$,
\begin{multline}\label{Gauss-HF-Spec-Value}
{}\_2F\_1\biggl(n+\frac{1}{2},n+1;n+\frac{3}{2};-1\biggr)
=\frac{(2n+1)!!}{(2n)!!}\frac{\pi}{4}\\
+\frac{2n+1}{2^{2n}}\sum\_{k=1}^{n} (-1)^{k} \binom{2n-k}{n} \frac{2^{k/2}}{k}\sin\frac{3k\pi}{4}.
\end{multline}


**My question is:** can one find an elementary function $f(t)$ such that
\begin{equation\*}
{}\_2F\_1\biggl(\frac{1}{2},\frac{1}{2};2;t\biggr)=f(t), \quad |t|\le1?
\end{equation\*}
In other words, is the Gauss hypergeometric series $\_2F\_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)$ an elementary function?


References


1. N. M. Temme, *Special Functions: An Introduction to Classical Functions of Mathematical Physics*, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996; available online at <http://dx.doi.org/10.1002/9781118032572>.
2. Feng Qi, Qing Zou, and Bai-Ni Guo, *The inverse of a triangular matrix and several identities of the Catalan numbers*, Applicable Analysis and Discrete Mathematics **13** (2019), no. 2, 518--541; available online at <https://doi.org/10.2298/AADM190118018Q>.
3. Feng Qi and Mark Daniel Ward, *Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function composited by inverse tangent, square root, and exponential functions*, arXiv (2022), available online at <https://arxiv.org/abs/2110.08576v2>.



Is the Gauss hypergeometric series ${}_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)$ an elementary function?
Maple does it in terms of complete elliptic integrals $\rm{K}$ and $\rm{E}$ ...
$$
{\mbox{$\_2$F$\_1$}\left(\frac12,\frac12;\,2;\,t\right)}={\frac {4\left( t-1 \right){\rm K} \left( 
\sqrt {t} \right) +4{\rm E} \left( 
\sqrt {t} \right) }{t\pi}}
\tag1$$
But that does not show it is **elementary**. In fact, I suspect it is not elementary.




---


Recall the known formulas:
$$
\rm{K}\big(\sqrt{t}\big) = \frac{\pi}{2}\;
{}\_2F\_1\left(\frac12 , \frac12 ; 1 ; t\right) ,
\\
\rm{E}\big(\sqrt{t}\big) = \frac{\pi}{2}\;
{}\_2F\_1\left(-\frac12 , \frac12 ; 1 ; t\right) .
$$
By themselves, they are not elementary. $(1)$ should follow from these two and a
contiguous formula for the hypergeometrics.
$$
{c\;\mbox{$\_2$F$\_1$}(a-1,b;\,c;\,x)}
 + \left( x-1 \right) c\;{\mbox{$\_2$F$\_1$}(a,b;\,c;\,x)}
 + \left( b-c \right) x\;{\mbox{$\_2$F$\_1$}(a,b;\,c+1;\,x)}
=0
$$


