Is the Gauss hypergeometric series ${}_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)$ an elementary function?

Question

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.

Answer 1

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. $$ {c\;\mbox{$_2$F$_1$}(a-1,b;\,c;\,x)} + \left( x-1 \right) c\;{\mbox{$_2$F$_1$}(a,b;\,c;\,x)} + \left( b-c \right) x\;{\mbox{$_2$F$_1$}(a,b;\,c+1;\,x)} =0 $$

Answer 2

That the function isn't presented in terms of elementary standard functions doesn't prove that it isn't an elementary function.

If the function is an elementary function, it's a Liouvillian function. I want to point out some perhaps applicable methods for looking if this function is an elementary function.

a)
We could check the known algorithms for solving recurrence equations.
We could check the known algorithms for symbolic summation, e.g. Gosper's algorithm.
see e.g. my answer at https://math.stackexchange.com/questions/1743544/are-there-some-techniques-which-can-be-used-to-show-that-a-sum-does-not-have-a/2329022#2329022

b)
$$_2F_1\left(\frac{1}{2},\frac{1}{2};2;t\right)=\frac{2}{\pi}\int_0^1\sqrt{\frac{1-\alpha}{\alpha-\alpha^2t}}d\alpha$$ We see, this is at least a Liouvillian function. We could check with Risch algorithm if $$\int\sqrt{\frac{1-\alpha}{\alpha-\alpha^2t}}d\alpha$$ is an elementary function.

Risch algorithm can decide if an indefinite integral is elementary or not. FRICAS has implemented Risch algorithm the furthest, but not yet completely. If a FRICAS result is elementary, we have an evidence. But if it is not obviously elementary, we therefore have to look further. see Abbasi, N. M.: Computer Algebra Independent Integration Tests

FRICAS 1.3.8:

integrate(((1-a)/(a-a^2*t))^(1/2),a)

(1)

   -
        6 t
     *
        weierstrassZeta  
              2
           4 t  - 4 t + 4
           --------------
                   2
                3 t

       ,
              3       2
           8 t  - 12 t  - 12 t + 8
           -----------------------
                        3
                    27 t
       ,
           weierstrassPInverse
                 2
              4 t  - 4 t + 4
              --------------
                      2
                   3 t
          ,
                 3       2
              8 t  - 12 t  - 12 t + 8
              -----------------------
                           3
                       27 t
          ,
              (3 a - 1)t - 1
              --------------
                    3 t
 +
     (- 4 t + 2)
  *
     weierstrassPInverse
           2
        4 t  - 4 t + 4
        --------------
                2
             3 t
    ,
           3       2
        8 t  - 12 t  - 12 t + 8
        -----------------------
                     3
                 27 t
    ,
        (3 a - 1)t - 1
        --------------
              3 t

/

      +-+  

    2 |1  
 3 t  |-  
    \ |t
                                     

Type: Union(Expression(Integer),...)

I don't know how to read the last part of this result.

It looks like the calculated antiderivative is not an elementary function, because the Weierstrass functions are not elementary functions. But give the arguments of the Weierstrass functions elementary functions?

We have to apply the integration limits now. Does that give an elementary function?

c)
Further approaches are the methods about elementary and Liouvillian integrals/solutions of linear differential equations of Davenport, Rosenlicht, Singer, Ulmer 1981 - 1992 and others.