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

- 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. - 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. - 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.

A new closed-form formula of the Gauss hypergeometric function at specific arguments, Axioms13(2024), no. 5, 24 pages; available online at researchgate.net/publication/380431446." $\endgroup$